REVIEWS OF GEOPHYSICS AND SPACE PHYSICS, VOL. 22, NO. 3, PAGES 275-308, AUGUST 1984
Gravity Wave Saturation in the Middle Atmosphere'
A Review of' Theory and Observations
DAVID
C. FRITTS
GeophysicalInstitute, University of Alaska, Fairbanks
This paper providesa review of recentadvancesin our understandingof gravity wave saturationin the
middle atmosphere.A brief discussionof thosestudiesleadingto the identificationof gravity wave effects
and their role in middle atmospheredynamicsis presentedfirst. This is followedby a simpledevelopment
of the linear saturationtheoryto illustratethe principaleffects.Recentextensions
to the linear saturation
theory, including quasi-linear,nonlinear, and transienteffects,are then described.Those studiesaddressing the role of gravity wave saturationin the mean circulation of the middle atmosphereare also
discussed.
Finally, observationsof gravity wavemotions,distribution,and variability and thosemeasurementsspecificallyaddressinggravity wave saturationare reviewed.
1.
INTRODUCTION
The term "saturation" was originally used to describethe
cessation of exponential growth of an instability due to
changesbrought about by its approachto finite amplitude.In
recent years, however, this term has also been applied to atmosphericgravity wavesthat are not unstablebut that experience an amplitude increasedue to the density decreasewith
height or a decreasein the intrinsic frequencyof the motion.
In this context, gravity wave saturation refersto any process
that acts to limit or reduce wave amplitudesdue to instabilities or interactionsarisingfrom large-amplitudemotions.Dissipation processessuch as molecular diffusion and radiative
cooling, in contrast, act more or lessindependentlyof wave
amplitude.
The saturation mechanismthrought to be most important
in the middle atmosphereis the turbulent breakdown of convectivelyunstableregionsproducedby the differentialadvection of more denseover lessdenseair by internal gravity wave
eling the middle atmospherecirculation and the recent•studies
of gravity wave saturation effects.
Studies of short-period fluctuations in the middle atmo-
spheredate back to the first observationsof traveling ionospheric disturbances(TID's) by Munro [1948, 1950] and
Beynon[1948]. Initial explanationsof these motions in terms
of atmospheric gravity waves were advanced by Martyn
[1950] and Hines [1955], but these theories, which relied on
gravity wave trapping and ionization resonance,respectively,
were later abandoned.Subsequentobservationsof TID's by
Munro [1953, 1958], Heisler [1958], and others showed them
to havephasevelocitiesof ,--50-200m s-•, wavelengths
of
severalhundred kilometers, periods ranging from 10 min to
severalhours, and a generaldownward motion. Also during
this time, information on atmosphericfluctuations in the 80to 115-km height range was obtained from observationsof the
motions and dispersionof meteor trails [Liller and Whipple,
1954; Greenhow,1959; Greenhowand Neufeld, 1959]. One example of the Liller and Whipple data is reproducedin Figure
motions. The turbulent diffusion that results from these convectiveinstabilitiesis assumedto reducethe wave amplitude 1. The meteor trail observationssuggestedmotions that were
to that value that just permits the continued generation of largelyhorizontal,with verticalscalesof 5-15 km, temporal
turbulence.Other mechanismsthat are likely to contributeto scalesof a few hours, and velocity and spatial scalesthat
gravity wave saturation under certain circumstancesinclude increased with height. With the benefit of both TID and
dynamicalinstabilitiesdue to large velocity shearswithin the meteor trail observations,Hines [1960] proposeda theory
wave field and nonlinear interactionsamong a spectrumof describingthe observedatmosphericfluctuationsas manifeslarge-amplitude waves. Each of these saturation mechanisms tationsof the (upward)propagationof internal gravity waves.
Since that time this theory has become almost universally
will be discussed
in moredetail in the followingsections.
Only in the last few yearshas gravity wave saturationbeen accepted and has provided a firm theoretical foundation for
recognizedto play an essentialrole in the large-scalecircu- subsequentmiddle atmospherestudies.Hines [1960] also anlation of the middle atmosphere.Until that time, althoughthe ticipatedsomeof the important effectsof gravity wavesin the
relevant gravity wave effectshad been anticipated,the large middleatmosphere,includingthe transportof energy,the genimbalance between the observedmiddle atmospherecircu- eration of turbulence, the likelihood of a nonlinear cascade of
wavesas the result of large gravity
lation and that required by the diabatic circulation in the energyto smaller-scale
wave
amplitudes,
and
the
possiblemodulation of the middle
absenceof a middle atmospheremomentum sourcewas not
atmosphere
response
due
to
the variable characteristicsand
appreciated.For this reason,studiesof gravity wave processes
and of the middle atmospherecirculationand structurepro- energiesof gravitywavespropagatingup from below.
Beginningin the late 1950's,a number of new techniques
ceededlargely in isolation until recently.The backgrounddiswere
employed to obtain additional data on the motions and
cussionpresentedhere reflectsthis parallel developmentof the
field. We begin by reviewingthe observationaland theoretical structureof the middle and upper atmosphere.Rocket vapor
studies of gravity waves in the middle atmosphere.This is trails and chemical releaseswere used to infer winds, wind
followed by a short discussionof the various efforts at mod- shears, and molecular and turbulent diffusion. The sodium
cloudexperimentsof Manring et al. [1959], Kochanski[1964],
and Rosenbergand Edwards [1964], for example,provided
Copyright 1984 by the AmericanGeophysicalUnion.
evidenceof gravity wave motions superimposedon mean and
tidal motionsto heightsof >•125 km. Similar techniqueswere
Paper number 4R0411.
0034-6853/84/004R-0411$15.00
used by Blarnont and de ,lager [1961], Zimmerman and
275
276
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
manner in which gravity wavespropagate in and interact with
a mean shearflow, includingthe production of turbulenceand
the acceleration of the mean flow due to a convergenceof
gravity wave momentum flux (see also Eliassen and Palm
E I00 [1960]). The latter effectwas subsequentlyrecognizedto occur
more generally wherever gravity waves are dissipated [Bretherton, 1969a, b] and thereby to provide a mechanismby
:z: 90which gravity waves may alter their environment and profoundly affect the circulation of the middle atmosphere[Bretherton, 1969c; Jones and Houghton, 1971, 1972; Breeding,
8O
1972; Hines, 1972; Lilly, 1972; Lindzen, 1973]. An important
I00
0
50
point in this context [Mcintyre, 1973, 1981] is that internal
WIND SPEED (m/s)
gravity waves do not possessmomentum but act only to
transport and redistributemomentumthat residesin the mean
Fig. 1. An example of a wind profile obtained from meteor trail
motions [after Liller and Whipple, 1954]. (Reprinted with permission flow. Nevertheless,the momentum source may be external to
of PergamonPress.)
the atmosphere,as in the caseof topographicforcing.
The convergenceof momentum flux due to dissipatingwave
motions was employed by Lindzen and Holton [1968] and
Champion [1963], Blamont and Barat [1967] and others to
Holton and Lindzen [1972] to explain the quasi-biennial oscildetermine turbulence structure and intensity up to the turbolation of the equatorial stratospherein the first demonstration
pause.These techniquesprovided estimatesof horizontal eddy of such an effect in the middle atmosphere. This theory was
diffusion in the mesosphereand lower thermosphereas large revisedand updated in the more recent work of Plumb [1977]
as 103-104 m2 s-1. More recently, aircraft and balloon and Plumb and McEwan [1978], but the underlying mechameasurementsand rocket vapor trail observationshave been nism, that of differential momentum flux convergenceby eastused to infer a vertical eddy diffusion in the stratosphere of erly and westerlywave motions,remainedthe same.A similar
•,0.01-0.2 m2 s-• [Lilly et al., 1974; Cadet,1977;Rosenberg mechanism was proposed by Dunkerton [1982c] to account
and Dewan, 1975].
for the observed semiannual oscillation of the equatorial
A rocket-borne grenade sounding technique was employed mesopause.A generaltheory of suchwave-mean flow interacby Theon et al. [1967] to measure the temperature structure tions has been advanced [Boyd, 1976; Andrewsand Mcintyre,
of the mesosphereand lower thermosphereas a function of
1976, 1978a], culminatingin the exact,finite-amplitudetheory
latitude and season. Pronounced latitudinal
and seasonal difof Andrewsand Mcintyre [1978b, c]. More recent discussions
ferences were observed, but there was, in most cases, evidence
of this theory and its applications have been presented by
of large-amplitude wave motions with superadiabatic lapse Mcintyre [1980] and Dunkerton[1980].
rates in the mesosphereand lower thermosphere.This strucIn parallel with the observational studies indicating the
ture is particularly evident in the data collected at Wallops presenceof wave-associatedsuperadiabatic lapse rates cited
Island during winter shown in Figure 2. Unstable lapse rates above, it was recognized that both gravity waves and tides
associatedwith large-amplitude wave motions were also obshould become dynamically and/or convectively unstable
served in the mesosphereby Groves [1966] and in the lower
above some level owing to their exponential growth with
thermosphereby Knudsenand Sharp [1965] usingan ion tem- height or approach to a critical level and that the ensuing
perature measurement technique. Additional indications of breakdown would likely result in the production of turbulence
wave motions and scaleswere provided by photographic stud- and of smaller-scale gravity waves [Hines, 1960; Hodges,
ies of noctilucent clouds (NLC) occurring at the high-latitude 1967; Lindzen, 1967, 1968a, b; Orlanski and Bryan, 1969; Orsummermesopause[Witt, 1962; Haurwitz and Fogle, 1969].
lanski, 1972; Geller et al., 1975]. On the assumption that convective instabilities within the wave field result in turbulent
This proliferation of wave and turbulence observations in
the middle atmospherestimulated considerabletheoretical atdiffusion and wave dissipation, Hodges [1969] calculated the
tention to wave and turbulence processes.The upward flux of eddy diffusioncoefficientneededto balancegravity wave amgravity wave energy due to sources in the troposphere was plitude growth with height. The eddy diffusion coefficientobaddressedby Gossard [1962]. Hines [1963] recognized the tained by Hodges as a function of horizontal and vertical
need for dynamical processesto account for the reversal of the wavelengthis illustrated in Figure 3. As is indicated, larger
meridional temperature gradient in the upper mesosphere, eddy diffusion coefficientsare required to balance the growth
though a specificmechanismwas not identified. The effectsof
of waves with greater vertical wavelengthsbecause of their
diffusion on the spectrum of upward propagating gravity
correspondinglylarger vertical group velocities.The valuesof
waves were examined by Pitteway and Hines [1963], and this eddy diffusion obtained by Hodges for typical wave paramediffusion was used by Hines [1965] to infer a dynamical heat- ters were in reasonableagreementwith those estimatesbased
ing in regions of wave dissipation.Other studiesby Friedman on observations of horizontal diffusion cited above. Hines
[1966], Midgley and Liemohn[1966], Hines and Reddy [1967],
[1970], in a related study, obtained an analytic expressionfor
and Lindzen [1970] addressedthe propagation, reflection,and
the eddy diffusioncoefficientin terms of the relevant gravity
filtering of gravity waves in the presenceof realistic temper- wave and mean flow parameters which agreed closely with
ature and/or velocity profiles.
that obtained by Hodges.
Also conducted at this time were a number of investigations
The laboratory and numerical studies by McEwan [1971,
that revealed the processesby which gravity waves are now
1973] and Orlanski [1972] showedregionsof convectiveinstathought largely to determine the large-scale circulation and
bility to precede the generation of turbulence and demonstructure of the middle atmosphere.The studiesby Bretherton strated the excitation of other motions due to convective in[1966] and Booker and Bretherton [1967] illustrated the
stabilitieswithin a standing wave. Delisi and Orlanski [1975]
I10 -
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
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Fig. 2. Rocket grenadetemperatureprofilesobtained at Wallops Island (38øN) during summerand winter from 1962
to 1965 [Theon et al., 1967]. Note the wavelike structuresand superadiabaticlapse rates. (Reprinted with permissionof
the American Meteorological Society.)
found similar effectsin a field of propagatinggravity wavesto
be consistentwith the phase speed condition for convective
instabilityof Orlanskiand Bryan [1969].
Using a different approach, Weinstock [1976] obtained a
nonlinear dispersionrelation which he used to infer the saturation of a gravity wave spectrumand the correspondingenhancedwave diffusiondue to strong nonlinear interactions.A
significantfeature of Weinstock'stheory was the prediction of
a highly anisotropic eddy diffusion with horizontal values
much larger than vertical values due to anisotropy of the
saturatinggravity wave field itself.
Other investigatorsattempted to show a need for an eddy
diffusion in the middle atmosphere on the basis of observed
distributionsof temperature or constituents.Johnsonand Wilkins [1965] estimated an upper limit on the mean eddy diffusion in the mesosphereand lower thermospherebased on the
required downward transport of heat. Their inferred vertical
diffusion,shownin Figure 4, was -,-10-100 times smaller than
observational estimates of horizontal diffusion. Colegroveet
al. [1966] examinedthe effectsof various eddy diffusioncoefficients on the composition of the lower thermosphere,and
Hodges [1970] invoked a constituent transport by largeamplitude gravity waves to account for the observeddistribution of helium in the lower thermosphere.Finally, the eddy
diffusioncoefficientneededto damp the diurnal tidal component at one location was calculatedby Zimmerman[1974] and
found to agree reasonablywell with the estimateprovided by
Johnsonand Wilkins. For the most part, however,thesecalculations yielded estimatesof vertical diffusion substantiallyless
than observationalestimatesof horizontal diffusion, lending
somesupportto the anisotropicdiffusionargumentsof Weinstock[ 1976].
Thus the gravity wave effectsnow thought to be crucial to
the large-scalecirculation and structure of the middle atmosphere,i.e., the momentum deposition and induced diffusion
accompanying the formation of convective instabilities by
large-amplitudegravity waves,were known over a decadeago,
but their importance went largely unrecognized until the
strengthof the diabatic circulationin the middle atmosphere
becameknown. Theseeffortsat understandingthe large-scale
circulationand structureof the middle atmosphereare briefly
described below.
The first calculationof net heatingratesin the middle atmospherewas that of Murgatroydand Goody[1958]. This study
and subsequentstudiesby Leovy [1964a], Kuhn and London
[1969], and othersshowedradiativeheatingand coolingto be
in approximatebalancethroughout much of the stratosphere
and portions of the mesosphere.Using the results of Murgatroyd and Goody, Murgatroyd and Singleton[1961] derived
the equilibrium meridional circulationneededto compensate
the net heatingrates.However,theseauthorsdid not attempt
to balancethe momentumbudget.Leovy [1964b] recognized
that dynamical processesmust account for the departure of
the atmospherefrom radiative equilibrium and simulatedtheir
effectsin his model of mesosphericcirculationby usinga uniform Rayleigh drag with a damping time of -,-15 days. This
provided a circulationand structurein qualitative agreement
with observations.
The zonally averagedsolsticezonal wind and temperature
fieldscompiledby Murgatroyd [1969] are shown in Figure 5.
Theseobservationsshow clearly the reversalof the meridional
temperature gradient and the closure of the mid-latitude jets
in the mesosphere.In order that the large-scalecirculation of
the middle atmosphereremain in approximate geostrophic
balance,of course,a reversalof the mean meridional temperature gradient must be accompaniedby a reversalof the vertical
shear
of the
mean
zonal
wind.
These
features
were
achieved in subsequentmodels of the middle atmosphere
circulation with the aid of a Rayleigh drag that increased
rapidly with height. Both Schoeberland Strobel [1978] and
Holton and Wehrbein[1980] useda Rayleighdrag with damping times that decreasedto •<2 days at the mesopause.This
278
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
•.
torques, and a correspondinglystronger zonal circulation,
than those obtained with the Newtonian cooling algorithm
used by $choeberland $trobel [1978] and Holton and Wehrbein[1980], suggestingthat eventhe large Rayleighdrag used
by theseauthors in the mesosphereis unable to provide the
requiredzonal deceleration.The summary of mean meridional
wind observationsby Nastrom et al. at Poker Flat, Alaska
(shown in Figure 8), and other sites provided important additional evidenceof a strongdiabaticcirculationin the middle
atmosphereand the need for an efficientzonal drag mecha-
io9
•
z=20K¾
nism. Additional
o
discussion of the need for and the effects of a
gravity wave drag in the middle atmosphereis providedin the
z
tutorial lecture by Geller [1983].
l0?
The nature of the drag mechanismneeded to balance the
thermal and momentum budgets of the middle atmosphere
was first examined by Houghton [1978] in his presidential
address to the Royal Meteorological Society. Houghton
argued that if planetary waves in either the winter or the
i
i
I i i I•L
summer hemisphere were controlling the mean meridional
circulation,then the summermesopausetemperature,which is
I0
I00
I000
particularly sensitiveto the strength of the mean meridional
HORIZONTAL WAVELENGTH Xx (KM)
circulation,should fluctuate in responseto the level of planetary wave activity. That such correlations were not found
Fig. 3. Eddy diffusion coefficientrequired to balancethe growth
of gravity wave amplitudeswith height in a motionlessatmosphere using either the Nimbus 6 pressuremodulated radiometer
data of Curtis et al. [1974] or the Nimbus 5 selectivechopper
[Hodges, 1969].
radiometerdata analyzedby Hirota [1975, 1976] led Houghton to concludethat planetary waves are not the primary vepermitted closure of the mesosphericjets and reversal of the hicle for the transport of momentum into the middle atmomean meridional temperature gradient, in better agreement sphere. Planetary waves can also be excluded from considerwith observations.The mean zonal wind and temperature ation becauseobservationsindicate (seeFigure 5) that mesocomputed by Schoeberl and Strobel for the northern hemi- sphericmomentum sourcesare requiredto reversethe vertical
spherewinter solsticeare shownin Figure 6. However, a Ray- shear of the mean zonal wind in both winter and summer
leigh drag of the form used by theseauthors can only act to hemispheres,whereasquasi-stationaryplanetary wavescannot
decelerate
the mean
zonal
wind
toward
zero
and
is thus
unable to accountfor zonal wind reversalsin the upper mesosphere and lower thermosphere.A Rayleigh drag also does
not accurately representthe effects of any relevant physical
mechanism. Furthermore, all of the above models of the
propagate through regions of easterly mean winds [Charney
and Drazin, 1961]. Houghton suggested,therefore,that gravity
wavesare the most likely candidatefor balancingthe thermal
and momentum budgetsof the middle atmospherebecauseof
their known sourcesin the lower atmosphere,their ability to
transport momentum vertically, and their convergenceof momentum flux in regionsof (turbulent)dissipation.It was subsequently argued by Holton and Wehrbein [1980] that their
Rayleigh drag profile could be thought of as a crude parameterization of wave drag due to the breakdown and dissipation
of vertically propagatinggravity waves.
middle atmospherecirculation usedNewtonian cooling to parameterize the effectsof radiative heating and cooling. The
study by Fels et al. [1980], however,showedNewtonian cooling to provide a serious underestimate of the net radiative
heating in the middle atmosphere. Thus any quantitative
agreement between observations and the results of these
middle atmospherecirculationmodelsseemslargely fortuitous
and may be attributed to the strengthof the assumedRayleigh
drag. Nevertheless,these models have servedto demonstrate
the importance of some zonal drag mechanismin accounting
IOO
for the large-scalewind and thermal structure of the middle
atmosphere.
9o
Additional indications of the need for a physical and efficient drag mechanismin the middle atmospherewere provided by the recent radiative heating calculationsof Allen et al.
[1979], Wehrbein and Leovy [1982], and Apruzese et al.
,• 7o
[1982] and the mean meridional wind observationsof Nastromet al. [1982]. The radiative heating calculationssuggested radiative relaxation rates substantiallyfaster than had previously been thought and, together with the large observed
departuresfrom radiative equilibrium, implied a correspondl0 s
;•
4 G 8 IC
ing increasein the strengthof the mean meridionalcirculation.
EDDY
DIFFUSIVITY
(CM•'S)
The radiative equilibrium temperature distribution, net radiative heating, and mean meridional and zonal winds calcuFig. 4. Variation with height of the eddy diffusionneededto aclated by Wehrbein and Leovy are shown in Figure 7. The count for the observedmesosphericand lower thermospherictemperstronger meridional circulation resulted in larger Coriolis ature structure[Johnsonand Wilkins, 1965].
-
--
FRITTS' GRAVITY WAVE SATURATIONIN THE MIDDLE ATMOSPHERE
'e,. '//
o
o
o o
279
280
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
126
-87.5
-58
a
-29
0
LATITUDE
19-87.5 -58
(DEG)
-29
b
0
29
58
8'/f5
LATITUDE
(DEG)
Fig. 6. Mean solstice(a) zonal wind and (b) temperaturedistributionsderived using a Rayleigh drag [Schoeberland
Strobel, 1978]. The meridional temperaturegradient in the mesosphere,while of the right sign, is very weak. Units are
degreesKelvin and metersper second.(Reprintedwith permissionof the AmericanMeteorologicalSociety.)
It was Lindzen [1981] who first proposeda simple scheme
by which both of the principal effectsof gravity wavesin the
middle atmospherecould be calculated. Like Hodges [1967,
1969], Lindzen consideredmonochromatic wave motions and
assumedthat wave amplitudeswould be limited by the formation of convective instabilities that would produce just that
level of turbulent diffusion required to maintain saturation
amplitudes.Lindzen also generalizedthe expressionfor turbulent diffusion obtained by Hines [1970] to apply to arbitrary
mean flows, including critical levels, and obtained a correspondingexpressionfor the mean flow accelerationsinduced
by wave dissipation and momentum flux convergence.The
latter, of course,is essentialin understandingthe role of gravity wave drag in balancing the thermal and momentum
budgets of the middle atmosphere. Becausethe mean flow
accelerationaccompanyingwave momentum flux convergence
drivesthe mean flow toward the phasespeedof the w•ave,the
saturation of gravity waveswith nonzero phasevelocitiesalso
offers an explanation for the mean zonal wind reversalsobservedin the uppermesosphereand lower thermosphere.
Following the study by Lindzen [1981] there has been a
resurgenceof interestin gravity wave saturationprocesses
and
effects.Dunkerton[1982a] and Weinstock[1982] recognized
showed that convectiveinstabilitiesshould dominate dynamical instabilities
for two-dimensional
of certain nonlinear
wave-wave
wave motions. The effects
interactions
on the saturation
of large-amplitude gravity waves were consideredby Weinstock [1982] and Lindzen and Forbes [1983]. Schoeberlet el.
[1984], Weinstock [1983], and Chao and Schoeberl[1984]
addressedthe wave and eddy transport of heat and constituents,arguing that gravity wave saturation should drive the
mean state toward an adiabatic lapserate, consistentwith the
viscous dissipation arguments presented by Walterscheid
[1981]. The evolution of the mean state accompanyinggravity
wave breakdown was studied by Walterscheid [1984] and
Dunkertonand Fritts [1984], and Lindzen [1984] and Schoeberl and Strobel [1984] consideredthe importance of various
gravity wave sourcesand the effectsof nonzonal saturation,
respectively.Someof the effectsof local, as opposedto global,
dissipation of saturating gravity waves were addressedby
Fritts and Dunkerton [1984].
Recentobservationalstudies,primarily usingvarious atmospheric radars, have also addressedgravity wave saturation
and its effectsin the middle atmosphere.These include momentum flux measurements[Vincent and Reid, 1983], rotary
spectralmeasurementsof low-frequencygravity waves [Vinthat saturation near a critical level results in a momentum flux
cent, 1984], the estimation of saturating wave parameters
divergencethat decaysas the critical level is approached,and [Fritts et al., 1984a], the inferenceof dynamical instabilities
Weinstock showed what the wave-induced diffusion coefficient
due to low-frequencywave motions [Balsley et al., 1983], and
observationsof layered turbulent structuresin the stratoand mean flow accelerationare related in a simpleway. Initial
modeling studies by Holton [1982] and Dunkerton [1982b]
sphere[Cadet, 1977; Sato and Woodman,1982a; Wand et al.,
revealedthat the saturationof gravity waveswith fairly simple 1983], among others. Such observationalstudies,however,
phasevelocity spectracan produce reasonableprofiles of the may be complicatedby the presenceof other quasi-horizontal
mean zonal wind in both steady state and transient forcing motions occurringat the same scales.Indeed, thesemotions,
configurations.The study by Matsuno [1982], while not inreferredto as two-dimensionalturbulence,have been suggestvoking gravity wave saturationexplicitly,incorporatedsimilar ed to accountfor a significantportion of the observedatmoeffects in that dissipation caused mean flow accelerations sphericvariability [Gage, 1979; Lilly and Petersen,1983; Natoward the appropriate gravity wave phase speeds.Finally, strom and Gage, 1983]. But becausevery little is known at
Holton [1983] and Miyahara [1984] incorporated the effects presentabout the relative importance of gravity waves and
of gravity wave saturation into models of the general circu- two-dimensionalturbulencein the atmosphereand because
lation of the middle atmosphere.
gravity waves are required to provide the middle atmosphere
Other studieshave addressedvarious aspectsof the gravity momentum source,we will consider only gravity wave prowave saturation problem. Fritts [1982] examined the interac- cesses
in this paper.
tion of a transient wave packet near a critical level and
In the following sectionswe review the recent theoretical
FRITTS'GRAVITYWAVESATURATION
IN THEMIDDLE.ATMOSPHERE
281
RRDIRTIVE
HERTING
90
80
•
'70
i.- 6o
_o 50
40
30
20
90
'75
60
45
30
15
0
-15-30-45-60-'75-90
LATITUDE
MERN ZONRL
liEFIN tiER I O [ ONFIL N [ NO
90
90
80
80
WIND
7o
• '70
• 60
• so
.1-50
w
40
'T' 40
30
20
c
LATITUDE
LATITUDE
Fig. 7. (a) The radiativeequilibrium
temperature
profile,(b)net radiativeheating,(c)meanmeridional
wind,and(d)
meanzonalwind obtainedwith an improvedradiativemodel[WehrbeinandLeovy,1982].Because
of the strongradiative
forcingandtherelatively
weakRayleigh
drag,themesospheric
jetsareabouttwicetoo strong.Unitsaredegrees
Kelvin
andmeterspersecond.
(Reprinted
withpermission
of theAmericanMeteorological
Society.)
and observationalstudiesrelating to gravity wave saturation
and its effects in the middle atmosphere.The various saturation theories,togetherwith their extensionsand limitations,
and the results of models incorporatingthe local effectsof
gravitywave saturationare describedin section2. Becauseit
is the simplestmodelconceptually,
the linear(monochromatic)
saturationtheory and its implicationsare reviewedin some
detail. Also discussedare the extensionsto the linear theory
that have been proposedto date, those saturationtheories
search.Those theoriesrelying upon convectiveor dynamical
instabilitieswithin the wave field to dissipatewave energy and
thus limit the primary wave amplitudes composeone group.
Theoriesin a secondgroup assumethat wave amplitudesare
limited, not by instabilitieswithin the wave field, but by nonlinear interactionsamong the componentsof the gravity wave
spectrum.Becausethere is observationalevidenceof both
typesof saturationin the middleatmosphere,
we revieweach
in this section.
that rely on nonlinearinteractionsamongcomponents
of the
wave field, and the likely effectsof localizedand/or transient 2.1. Linear Saturation Theory
saturation events.The mechanisticmodels incorporating grav-
ity wave effectsin the large-scalecirculationof the middle
atmosphereare discussed
in section3. These range in complexity from one-dimensional
wave-mean flow interaction
modelsto three-dimensionalprimitive equation modelsthat
permita rangeof wavemotionsand responses.
In section4 we
reviewwhat little is known of gravity wave characteristics
and
climatologyin the middle atmosphereas well as thoseobservations addressinggravity wave saturation specifically.Processesaffectingthe distributionof gravity wavesin the middle
atmosphereare discussed
in section5, and the conclusions
are
presentedin section6.
2.
GRAVITY
WAVE SATURATION THEORY
There have been a number of theoretical
studies over the
last 2 decadesthat have addressedvarious aspectsof the grav-
The gravitywavesaturationtheorythat is the simplestconceptuallyis one that relies on instabilitiesin a wave field
assumedto be describedby linear wave dynamics.This theory
was first advanced by Hedges [1967] to explain the occurrenceof turbulencein the middle atmosphere.Subsequent-
ly, Hedges[1969] and Hines [1970] calculatedthe level of
turbulent diffusionneededto balancewave growth with height
in an unshearedenvironment. It was Lindzen [1981], however,
who first attempted to describe,in a quantitative manner,
both of the principaleffectsof gravity wave saturationnow
thought to be important in the middle atmosphere.This
theory is outlined below.
We consider first, in the absence of saturation, adiabatic
inviscid gravity wave motions with intrinsic frequenciesto
such that
ity wavesaturationproblem.Early studiesaddressed
the turbulent diffusion neededto accountfor wave dissipation,while
subsequent
effortsexaminedboth decelerationand diffusionas
well as departuresfrom simplelinear theory.However,all of
these studies have followed one of two distinct lines of re-
f • to • N
(1)
in a mean state atmospherein hydrostatic balance. Here
f- 20 sin •b is the inertial frequency and N is the
282
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
With the above assumptionsthe x and z momentumequations,the adiabaticenergyequation,and the continuityequation may be combinedto yield
Poker Flat Summer
Mean Meridional
Winds
i
I
I
t Groves
(1969)
km
%: + t•- c)• (t•- c)
(a- c)H 4H•
(6)
I
where c is the horizontal phasevelocity of the wave motion.
This equation,known as the Taylor-Goldsteinequation,describesthe amplitude and vertical structure of the vertical
perturbationvelocityof atmosphericgravitywavesin an environment with variable N and if. For our purposes,however,it
is reasonable
to assumethat N: is constant,that if:: is small,
and that H is large. Also, from (1) with co= k(c- •) we see
x
88
that k: may itselfbe neglected
as a first approximation
(the
wave motions themselves are approximately hydrostatic).
Thus (6) becomes
8•
w::' + [Ne/(a - c)2]w' = 0
(7)
This equation has the approximate WKB solution I-Bender
and Orszag, 1978'!
w'(z)= Am- •/:eism:'
80-
I
I
I
-50
-20
-10
(8)
where rn = N/(ti- c) describesthe variation of the vertical
wave number with N and tZ Provided that rn is slowlyvarying,
the verticalperturbationvelocityhasthe form
0 ms-1
1979 June 9- July 16
1980 June 2:1-July 2:4
w'(x,z, t) = Am-1/2e:/2Hei(kx+mz-kcO
1981 June15- July I0
Fig. 8. Mean meridional wind near the mesopauseat Poker Flat,
Alaska (65øN), for three summers[-Nastrornet al., 1982].
(9)
Other perturbationquantitiesare obtainedin terms of w'
from the linearizedequationsof motion. From the continuity
equationwe seethat for m slowlyvarying,
u' = -mw'/k
Brunt-Viiisiiliifrequencydefinedby
(10)
and from the adiabaticenergyequationwe find that
N•(z)
= • •zz+
- ••
(•)
o'= [- 0•/#,(•- c)]w'
(11)
where•' and 0 are the meantemperatureand the potential It is not the purpose of this paper, however, to review the
temperature,respectively.Then to a good approximation,inertial effectsare unimportant and motionsmay be taken to be
two-dimensional. Without loss of generality we may also
assumemotions to occur in an x-z plane with a mean velocity
fi(z) in the x direction. Finally, we assumethat all field variables may be written as the sum of a horizontal mean and a
perturbation about the mean,
theory of atmospheric gravity waves. For additional discussion of their
structure
and characteristics
the interested
readeris referredto the works by Hines [1960], Eckart ['1960],
Tolstoy [1963], Georges[1967'], Yeh and Liu [1974], and Gossardand Hooke [1975].
Differentiating(11) with respectto z and assumingthat perturbation quantitiescontribute most to the vertical variations,
we obtain
,p(x,z, t)= ,•(z) + ,p'(x,z, t)
(3)
0/-
with
•(z)=•
0dx
(5)
The quantities k and rn are the horizontal and vertical wave
numbers,respectively,of the perturbation,,•nis the horizontal
wavelength(,• = 2•/k), and H is the local atmosphericscale
height(H = R•'/g). The factore:/2His just that requiredto
offset the density decay with height and insure that energy is
conserved until saturation
occurs.
mw'
(a- c) /½
-
u't•:
(a- c)
(12)
(4) Now the atmosphereis staticallyunstablewhenever
that productsof perturbationquantitiesmay be neglectedin
the equations of motion (this is a poor approximation for
saturatinggravity waves),and that eachperturbationquantity
may be representedas a singleFourier component,
q/(x, z, t) = q/(z)e:/:He
ik(x-cO
-0:
(0 + 0'): < 0
(13)
(T + T'): < -a/cp
(14)
or, from (2), where
But with (12) we see that theseconditionscorrespond,in the
case of monochromatic wave motions, to a simple condition
on the horizontal perturbationvelocity,
u' > c - a
(15)
Thus for monochromaticmotions,occurrencesof convectively
unstableregions within the wave field correspondto regions
in which the total (Eulerian) parcel velocity, ff + u', exceeds
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
283
the horizontal phase speed c of the wave. A more general
expressionof this correspondencein terms of an "exchange"
velocity was provided by Orlanski and Bryan [1969] for the
case of a superpositionof waves. Nevertheless,such regions
are most unambiguouslyidentified in the wave-associated
temperature(or potential temperature)field becauseno prior
knowledgeof wave structureis required.
The basic premiseof the linear saturation theory advanced
by Hodges [1967, 1969] and Lindzen [1967, 1968a, 1981] is
that convectivelyunstableregionsappearingwithin the wave
field result in the production of turbulenceand just that level
of eddy diffusion necessaryto restrain wave amplitudes to
valuesnear neutral static stability. Implicit in this formulation
is the assumption that saturation does not otherwise affect
wave propagation or characteristics.If we assume that the
effectsof turbulent diffusion are immediate and complete,the
saturation
conditions
become
o
Oz'_• -Oz
or Tz'_• - •z - g/Cp
(16)
and for monochromatic wave motions,
lu'l _• Ic - al
(• 7)
The saturation conditions place no constraints on smallamplitude wave motions. However, wave motions with fluctuations that exceedsaturationvaluesare subjectedto turbulent
dissipationthat limits wave amplitudesand producesa vertical divergenceof the gravity wave momentum flux, which is
nondivergentfor steady,conservativewave motions.These effects are illustrated in Figure 9. The turbulent dissipationof
saturatinggravity waveswas suggestedby Hodges [1969] to
account
for the observed level of turbulent
diffusion
of the turbulent
diffusion
Fig. 9. Schematic of the growth with height and saturation of a
gravity wave due to convectiveinstability. Wave damping produces
both a divergenceof the vertical flux of horizontal momentum and an
accelerationof the mean flow toward the phase speedof the wave.
Decelerationand diffusionceaseabovethe criticallevel(z = zc)in the
linear theory.
The momentum flux associatedwith the saturated gravity
wave is then
! ! •0 !
POtts Ws = •
Us ws -- •
Po k
•
2 N
(d- c)3
(21)
Clearly, the momentum flux above zsreflectsvariations in z
both of the density, po(Z),and of the mean velocity in the
direction of wave motion, •(z). Thus the induced mean flow
accelerationgivenby (18) above Zsis the sum of two terms,
and mean flow acceler-
ation accompanyinglinear gravity wave saturation.
Following Lindzen [1981], we assumethat gravity waves
propagateupward from sourceregionsin the troposphereor
lower stratospherein a conservativefashion until they achieve
saturationamplitudes.Thereafter,we assumethat wave amplitudes are limited by the saturation conditions (16) and (17).
The phasevelocityspectraand the regionsof saturationenvisaged by Lindzen for winter and summer at mid-latitudes are
illustrated in Figure 10. Below the level at which waves saturate, denotedZs,the mean flow acceleration,
a, = --(PoU'W')z/Po
o
in the
middle atmosphere. Momentum deposition accompanying
gravity wave dissipationwas suggestedby Houghton [1978]
and Lindzen [1981] to provide the momentum sourceneeded
to satisfythe thermal and momentum budgetsof the middle
atmosphere.In the remainder of this section we outline the
derivations
o
(18)
• =
2N
H
--k(ff
--c)2
I(ff
--c)3az] (22)
In an environment where •-
c varies little, the induced mean
flow accelerationis approximatelyconstantwith height,
Us'Ws' - k(a- c)3
a• • •
The induced acceleration
H
=
(23)
2NH
in a sheared environment
is not con-
stant but increasesor decreases
with the intrinsicphasevelocity, ff- c. In particular, the induced mean flow accelerationis
proportionalto (•- c)2 as a criticallevel (where• = c) is
approached [Weinstock, 1982; Dunkerton, 1982a]. The form
is zero becausethe gravity wave momentum flux is constant
away from critical levels[Eliassenand Palm, 1960]. The saturation level zs may be inferred from the wave and mean flow
parametersspecifiedin the lower atmosphereby using the
saturationconditions[Holton, 1982; Lindzen,1984]. Above zs
the saturated horizontal perturbation velocity (assumingthat
c - ff doesnot increase
fasterthan ez/•-n)is
lus'l= Ic- •1
(19)
and from (9) the saturatedvertical perturbationvelocityunder
our assumptionsis
k
k
'= -- mUs'= • 02- c)2
Ws
(20)
and direction
of the mean
flow
acceleration
are shown
in
Figure 9 with the dashedmean velocity profile. Note that the
tendency is always to accelerate the mean flow toward the
phasespeedof the gravity wave. Thus the saturation of gravity waveswith positiveand negativezonal phase speeds,respectively,provides a mechanismby which the summer and
winter mean zonal flows in the upper mesospheremay be
reversed, in line with recent observations. It should be
stressed,however, that the induced acceleration is not limited
to the zonal direction but is applied in whatever direction the
wave is propagating in relation to the local mean flow.
We now turn to a determination of the turbulent (or eddy)
diffusioncoefficientneededto just maintain the gravity wave
at saturation amplitude. The departure from conservative
284
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
needed to conserve wave energy (or more correctly, wave
action) in the absenceof dissipation;the secondterm arises
from the constraint
6O
_
Zbreok
_
WAVES
on u' due to the variation
_
D•
III
,a)
<From
[øst
' ,
J , ILL"
100
50
speeds'
,
50
lOO
!
T
I
!
I
!
(27)
(28)
But this can be rewritten by defining
sPEEO ( msec'1)
I
0'' ----m'-D,
ik(a- c)O'+ w'O:= -m2DO'
l
0
c with
This is equivalent to the assumptionof a turbulent Prandtl
number of unity. Thus the energy equation becomes,for example,
', 'Pr0htbtfe.d
ph0se
[ I //
of a-
height. The variation of a- c contributes additional dissipation if la - cl is decreasingand partially offsetsthe term due
to densitydecayif I• - cl is increasingwith height.
Again following Lindzen [1981], we assumethat the wave
amplitude decreaseaccompanyingsaturation is accomplished
via a dissipationterm in both the momentum and the energy
equationsof the form
kci---mZD
(29)
ik(a- e)O'+ w'O:= 0
(30)
I
as
80
//
A
4
Geostrophlc
/
70
60
[
Ju.ly//A
]
-
x
Rewritingthe momentumequationsin the samemanner,we
seethat this correspondsto a redefinitionof the vertical wave
number(now complex)in terms of • as
o-•
m = m, + im• = N/(a-
d)
(31)
Then the imaginary part of the vertical wave number and the
required eddy diffusion are related as follows
•0
,RN • o
_
x Wollops
1961-65
•
D•
Wollops
erenodes
20o•oh•bded
Green
R1964-65•
ASummer
1960-64
phose,
•
speeds_ •
•
o
5o
N
2]-/
D• '••
k(a-cp[
1 2(a-c)
3 fi:]
lOO
Fig. 10. Schematicillustrating the allowed and prohibited phase
speedsfor gravity waves at Wallops Island for winter and summer
[Lindzen, 1981]. Note that the summer wind profile preventsstationary wavesfrom enteringthe middle atmosphere.
growthcan be representedas an imaginarycontributionto the
vertical wave number above
%'(z)= w'(z)e-='(=-=')
(24)
This may be evaluated for rn• in terms of the saturated and
unsaturatedperturbation velocitiesas
mi=•z 1+
I r&w'(z•)
w,(z,7
L
(32)
,j
(33)
In the absenceof mean wind shear this expressionhas the
same form as the eddy diffusion coefficient obtained by
Hodaes [1969] and Hines [1970]. Finally, by comparing(22)
and (33) we find that the mean flow accelerationinduced by
saturation and the eddy diffusion responsiblefor wave dissipation are related in a very simpleway [Weinstock,1982],
a, = [N2/(c- a)]D
(34)
As is pointed out by Lindzen [1981], however,the two effects
addressedby this simple linear model, inducedmean flow accelerationand enhanceddiffusion,while related, are separate
manifestationsof gravitywavesaturation.
We now estimate,using (22) and (33), the decelerationand
eddydiffusionimplied by this linear theory for a gravity wave
with typical (observable)parameters.We assumea uniform
mean flow for simplicity, a scale height H= 6 km, a
Brunt-V•is•l• frequency
N = 0.02 s-•, an intrinsicphasevelocity c- ff = 30 m s-•, and a horizontalwavelength• =
(25)200 km. The resultingdecelerationand diffusionare
Using (9) and (20), this becomes
m,=
3
or using(26),
SPEEO(msec'1)
1
mr
'
10
(b)ß From
Eosf
/' FromWesfß 1oo
km•(a- c)
a• = 305m s-• d-•
(35)
D = 265m2 s- •
(36)
3
- (a- c)
(26) and
The first term is due to departurefrom the exponentialgrowth
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
The magnitude of the decelerationis somewhat larger than
what is thought to be needed to balance the zonal accelerations produced by Coriolis torques due to the mean meridional circulation. The value of eddy diffusion is consistent
•tltll
(7(:1,111{71
--':'•---': .....[;3LIIIIO.,LKJ3
:.....
of the required vertical diffusion.From
the expressionsfor fi, and D, however,it is clear that both
quantitiesare stronglyinfluencedby the intrinsicphasevelocity of the wave motion, c- fl. This is easily understoodby
recognizingthat the vertical group velocity, for those wave
motionsconsideredhere,is itselfa strongfunction of c - •,
&o
kN
k(•-c) 2
Ca:
= 2rrl
-- m2--•
MEAN
285
WIND
10.0
9.0
8.0
7.0
6.0
(37)
Thus the rate at which wave energyis suppliedto the region
of saturation dependsstrongly on c- fl. The dependenceon
3.0
horizontal wavelength is of somewhat lesser importance.
However, this dependencedoes imply that the momentum
2.0
transport, if accomplishedby wave motions with small horizontal wavelengths,need only be intermittent.
The linear saturation theory presentedabove clearly has a
number of shortcomings.For example, the neglectof various
effects,including wave superpositionand interaction, wave
0.0
0.0
2.0
t}.0
6.0
8.0
10.0
12.0
lt}.0
16.0
transienceand horizontal localization, quasi-linearmean flow
TIME
accelerations,and the detailednature of the saturationprocess
Fig. 12. A time-heightcrosssectionof the inducedmean wind for
may not be justified in general.Nevertheless,the linear theory
has establisheda basic understandingof gravity wave satu- the saturated slowly varying wave action model [Coy, 1983]. The
nondimensionalmean wind undergoessubstantialaccelerationsdue
ration and its effectsupon which subsequentstudiescan build. to self-acceleration
of the gravity wave.
2.2. Extensionsof the Linear Saturation Theory
In the wake of the paper by Lindzen[1981] there have been
numerousstudiesaimed at implementing,extending,or modifying the linear saturationtheory.The studiesperformedup to
this point have addresseda number of important issues,including wave transience, quasi-linear effects of wave-mean
flow interaction, effectsof local convectiveinstability, and
consequencesof nonzonal dissipation, among others. There
remain,however,a number of equallyimportant, and in many
respectsmore challenging,topics. The principal findings of
thosestudiesperformedto date are summarizedbelow.
As an extensionof his earlier work, Dunkerton[1982a] applied the saturation parameterization suggestedby Lindzen
[1981] to his investigationof wave transiencein a compressible atmosphere.Togetherwith the assumptionof a constant
gravity wave horizontal phasevelocity,this resultedin a nondimensionalwave action equation that predicted quantitatively differentbehaviordue to saturationeffects.This equation (seeDunkerton for scalings)may be written
(aA/at) + [(a/az) - 1]B = 0
(38)
where A is wave action density and B is wave action flux,
defined as the product of A and the vertical group velocity
(1 - •)•-. Under the saturationhypothesis,
however,the wave
action densityis constrainedto be
As= «(1 - t•)
(39)
causing different saturated and unsaturated wave action
fluxes.Dunkerton investigatedthe consequences
of saturation
with the wave action equation,usingthe method of characteristics. The characteristics
5
I0
15
TIME
Fig. 11. Characteristicsfor a saturated wave solution of the analytic, quasi-linearwave action model [Dunkerton,1982a]. Relative to
the unsaturatedsolutions,the saturatedmodel predictsa more rapid
internal shock descentand a more gradual approachof the nondimensionalmeanflow to the (nonaccelerated)
phasespeedof the wave
motion. (Reprintedwith permissionof the American Meteorological
Society.)
for one saturated
wave case are
shownin Figure 11. The lower solidcurvein the Figure representsthe locationof the internal shockanticipatedby Dunkerton [1981]. At the point where the wave action attains the
saturatedvalue As = 1/3, however, the characteristicslope
changes,reflectingthe reducedwave action flux above. Also
changed,thereafter,is the rate at which the mean flow above
the saturation level approachesits limiting value t2= 1. Becauseaccelerationsat upper levels are reduced by the saturation condition, those at lower levels must increase. Thus the
principal effectsof saturation in this quasi-linearmodel are a
more rapid internal shock descentand a more gradual mean
flow accelerationat upper levels due to the limit on wave
amplitude.
One effect, not addressedby Dunkerton [1981, 1982a],
286
FRITTS' GRAVITY WAVE ,SATURATION IN THE MIDDLE ATMOSPHERE
10.0
•
I
•
I
I
I
•
t
i
formedby Fritts [1982] demonstratedthat the regionswithin
which dynamical(Ri < 1/4) and convective(Ri < 0) instabil-
I
ities can occur, where the Richardson number is defined as
9.0
Ri =
8.0
7.0
6.0
4.0
3.0
2.0
1.O
0.0
0.0
.2
.•t
.6
.8
1.0
NERN
1.2
1.q
1.6
1.8
2.0
NINO
Fig. 13. Accelerationof the nondimensionalmean wind following
switch-on of a gravity wave with an initial critical level at 1.0 [Coy,
1983]. The variousprofilesare labeledwith nondimensionaltime.
which is potentially very important for gravity waves in the
middle atmosphere is the self-accelerationof gravity waves
due to their induced mean flow modifications.
This is a mech-
anism by which gravity wavesmay appear in the stratosphere
or mesospherewith horizontal phase velocitiessubstantially
different
from those that characterized
their motions
at lower
levels.This processwas investigatedby Coy [1983] for both
shearedand unshearedmean flows, using a generalizationof
the wave action equation of Dunkerton. Solving the wave
action equation under the saturation assumptionnumerically,
Coy obtained resultsin the absenceof an initial mean shearin
qualitative agreement with those of Dunkerton [1982al at
lower levels. At upper levels, however, the (nondimensional)
mean flow (see Figure 12) exhibited accelerationsof up to 3
times the initial gravity wave phase velocity. Because this
study incorporated the slowly varying approximation, critical
levelswere not a possiblefeature of simulationsthat contained
none initially. Thus the mean flow accelerationsobservedin
Figure 12 imply an increaseof more than 200% in the gravity
wave phasevelocity in portions of the model domain.
A similarconclusionis suggested
in the presenceof a mean
shear by the results [Coy, 1983] shown in Figure 13. In this
casea gravity wave with an initial phase velocity c = 1 that
was "switched"on at t--0 approachesa critical level where
a-- 1. But while there is evidencethat the vertical group velocity, and thereforec - a, is decreasingas the evolution proceeds,the phase velocity (c > t•) at the leading edge of the
wave field already exceedsthe mean velocity at the critical
level. Nevertheless,large self-accelerationsappear to be confined below the critical level in the slowly varying model. Additional evidenceof gravity wave self-accelerationis provided
by the resultsof Fritts and Dunkerton [1984-1,discussedlater
in this section.
Other studiesof gravity wave propagation and breakdown
have offered insightsinto the nature of the saturation mechanism. The transient critical-level interaction simulations per-
N 2 (Q/O)(c•/C•z)(O
+ 0')
u:2 [(•/•z)(a + u')]2
(40)
are very nearly coincidentand appear almost simultaneously
for two-dimensionalgravity wave motions.This suggeststhat
convectiveinstabilitiesmay appear and dissipateexcesswave
energybeforedynamicalinstabilitiescan achievelarge amplitudes. Support for this conclusionis provided both by the
nature of the nonlinear gravity wave breakdown modeled by
Fritts (seeFigure 14) and the laboratory observationsof gravity wave-meanflow interactionby Koop [1981] and Koop and
McGee [1984], an exampleof which is shownin Figure 15. In
each case there are indicationsof overturning motions with
aspectratios of approximatelyunity. Dynamical instabilities,
on the other hand, typically have aspectratios of •<7. Koop
and McGee were also able to observethe dynamical instability of a two-dimensionalinternal gravity wave field. However, they state that convectiveinstabilities are much more
easilygenerated.This is becausethe range of amplitudesfor a
particular wave motion that supportsdynamical instabilities
without simultaneouslypermitting convectiveinstabilitiesis
very small.
The situation is somewhat different for gravity wave motions with intrinsicfrequenciesnear the inertial frequencyf.In
this case,rotational effectsenhancethe perturbation velocity
shears relative to the perturbation temperatures, causing
larger spacingsbetweenthe regionswithin which dynamical
1.0 0
,
2, 1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
z/zø 0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0
•
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
z/zø 0.5
0.5
0.4
0.4
0.3
).3
0.2
(b)
.:•
0.1
0.1
0.0
0.0
0
,
Fig. 14. Onset of a convectiveinstability due to the approachof a
gravitywaveto a criticallevel [Fritts, 1982]. Timesare (a) 2.0 and (b)
2.4 wave periodsafter excitation.
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
287
Fig. 15. Shadowgraphof the convectiveinstability of an internal gravity wave causedby a decreaseof the intrinsic
wave frequency[Koop and McGee, 1984]. Note the similaritiesbetweenthis flow and that shown in Figure 14. (Reprinted
with permissionof CambridgeUniversity Press.)
and convective instabilities are possible.Together with the
smallervertical group velocitiesof suchmotions,this suggests
that dynamical instabilities,when they occur, should be associated preferentially with the saturation of wave motions
with low intrinsic frequencies.This mechanism was invoked
by Balsley et al. [1983] to account for a number of observations of wave motions and apparent wave saturation near the
high-latitudesummermesopause.
The first study incorporatingeddy diffusiondue to gravity
wave saturation in the thermodynamicenergy equation was
that by $choeberl et al. [1983]. For various mean velocity
profiles these authors determined the eddy diffusion profile
neededto just maintain saturated wave amplitudes and then
inferredthe resultinginstantaneousdecelerationand net heating profiles. With their assumption of a uniform eddy diffusion, Schoeberl et al. observed that both turbulent fluxes and
wave fluxes (nonzero owing to wave dissipation)of heat contributed to a downward heat transport.A heating at all levels
of turbulent dissipation was also inferred. The net heating
determined by Schoeberl et al. for canonical summer and
winter mid-latitude zonal wind profiles(scaledby zonal wave
number) is illustrated in Figure 16. In each casethe turbulent
heating is not sufficientto offset the cooling at upper levels
brought about by the induced downward turbulent and waveassociatedheat fluxes.At lower levels,both turbulent heating
and heat flux divergencescontributed to a net heating, thus
driving the mean lapse rate toward adiabatic. Likewise, a
downward heat flux due to wave motions undergoingbroad
spectrumsaturation was inferred by Weinstock[1983]. Schoeberl et al. also examined the effects of radiative cooling or
heating on inertio-gravity wave motions and concludedthat
suchmotions with intrinsicphasevelocities
Ic -- •21•< 20 m s-•
(41)
will be radiatively damped sufficientlyrapidly that saturation
will not occur. Radiative damping was also considered by
Tanaka [1982] and shown to be important for small intrinsic
phasevelocitiesand large horizontal wavelengths.
In a related study, Walterscheid[1984] used a quasi-linear
wave-mean
flow interaction
model
to examine
the mean flow
modifications accompanyinggravity wave saturation near a
critical level. Like Dunkerton [1982a], Walterscheid assumed
that the gravity wave phase velocity remained constant.This
led to a rapid decreasein the eddy diffusioncoefficientneeded
to damp the wave motion becauseof the induced momentum
flux convergenceand the resulting decreaseof the intrinsic
phase velocity, c- if. Owing to wave transience,Walterscheid
observed gravity waves to saturate first near their critical
levels.As in the study of Schoeberlet al., the heat fluxes,due
both to the dissipatingwave motion and to the horizontally
uniform eddy diffusionacting on the mean temperaturegradient, caused the mean flow to tend toward adiabatic. Walters-
cheid suggestedthat the observed mean flow modifications
could result in the dynamical instability of the mean state
several hours after the onset of saturation.
A major modificationto the linear theory advancedby Lindzen [1981] was suggestedby Chao and $choeberl [1984].
These authors argued that becauseturbulencefirst appearsin
regionsfor which
(42)
whatever local mixing occurswill not appreciablyreduce the
288
FRITTS.' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
r'øl
'
'
I
I
I
I
/ i/11
/ .d.I I
,'/•
I I
I
I
I
•
I
r.,III
/11
//
/
/
/ /
/
/
//
//
I ....
I
'
NETHEATING
EX, 4
,^.^ME'rE.I-
Fritts and Dunkerton [1984] addressedthe more general
problem of saturationdue to the exponentialgrowth of wave
amplitude with height. Consistent with the wave action
models of Dunkerton [1982a] and Coy [1983], this study
NET HEATING
,:x,,,
I .......PARAMETERI-
I
I'---_
.8-
o
-15
-10
-6
0
5
10
x10
3kmdegd•y-•
No
N
Fig. 16. Net heating in regions of gravity wave saturation for
canonical summer and winter mean wind profiles [Schoeberlet al.,
-r
1983].The net heatingis a productof turbulentheatingand both
turbulent
andwavetransports,
The heatingis scaledby the zonal
tu
wavelengthof the gravity wave.
amplitudeof the wave in the temperaturefield. Thus any wave
dissipationmust be accomplishedthrough the eddy diffusion
term in the momentum equations. This implies an eddy
Prandtl number Pr >> 1. Becauseeddy diffusionwas assumed
to play an equal role in the momentum and thermodynamic
energy equations (Pr-1)
in Lindzen's formulation, this
theory of gravity wave saturation requiresdouble the level of
eddy diffusion predicted by Lindzen to provide the same
degreeof wave dissipation.Chao and Schoeberlalsosuggested
that the net flux of heat due to wave and turbulent
01
o
•
t
•
t
•
•r
LONGITUDE
---,-
motions
should remain more or less unchanged becausethe loss of
wave heat fluxes would be compensatedby the larger turbulent fluxes. However, this study did not addressthe implicationsof a localizedturbulent diffusionfor the transport of
heat. Additional studieshave indicated that an eddy Prandtl
numberPr ,-• 20 may be appropriate(Schoeberl,private communication, 1983).
The studiesby Dunkertonand Fritts [1984] and Fritts and
Dunkerton [1984] both employed a relaxational convective
adjustmentschemeto accountfor the dissipationof saturating
.6gravity waves.The former study focusedon the wave-mean
flow interactionand saturationeffectsin the vicinity of a critical level.The effectsof saturationin the convectiveadjustment
model were found to be in good agreementwith the predictions of the semianalytictheory of Grimshaw[1975], Dunkerton [1982a], and Coy [1983]. These effectsincluded a modified
momentum flux convergencedue to saturation and the formation of internal and trailing shocks.The internal shock, in
particular, was found to be significantbecauseit resultedin
.2the partial reflectionof the trailing wave packet. These authors also found local convectiveadjustmentto preservethe
grossfeaturesof the saturatingwave motion, suggesting
that
local turbulent diffusionmay not disrupt the wave field substantially.A comparisonof the unsaturatedand locallyadjusto
o
ed wave fieldsis displayedin Figure 17. Clearly, the principal
result of local dissipationis the attenuation of the incident
LONGITUDE
-.wave packet. Nevertheless,local convectiveadjustmentwas
Fig.
17.
Comparison
of
(top)
an
unsaturated
and (bottom)a satuobservedto permit the excitation of harmonicsof the fundarated gravity wave as a critical level (at z/zo -0.6) is approached
mental wave motion which were able to propagateinto the [Dunkerton and Fritts, 1984]. Note that local saturation does not
region above the critical level.
substantiallydisrupt the wave field.
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
1.0
i
i
i
289
i
i
-40
-20
0
20
40
-40
-20
0
20
40
-20
0
U'1(ms-1)
u•I (ms-1)
20
U•I (ms-1)
Fig. 18. Horizontal velocity perturbation at (a) t = 4.5, (b) 5.0, and (c) 5.5 (x 1768 s) for a saturatedgravity wave
exhibitingself-acceleration
and critical-leveldislocationdue to large amplitude[Fritts and Dunkerton,1984]. The initial
criticallevelwaslocatedat z/zo = 0.6.The intrinsicphasespeedis shownby a dashedline.
found
mean
flow
accelerations
due to wave
transience
to
permit saturation at a lower level than that predictedby linear
theory. Such effectsmay help explain the occurrenceof saturation at lower levels that those anticipated on the basis of
linear theory [Lindzen, 1984]. Fritts and Dunkerton also
found local convective adjustment to impose a limit on the
horizontal perturbation velocity above the level of saturation
consistentwith that predictedby linear saturation theory. The
most significantresult of this study,however,was the observation that saturating gravity waves can undergo substantial
self-accelerations,producing phase velocities in the mesospherevery different from thosecharacterizingthesewave motions in the lower atmosphereand causingcritical levelsto be
displacedupward by large distances.Thus wave packets of
sufficient amplitude and horizontal extent may propagate
beyondthe height of their original critical level, encountering
a new critical level wherethe acceleratedintrinsicphasevelocity is zero. This critical-leveldislocationis illustratedin Figure
18. The original critical level,basedon the initial phasevelocity, was located at Z/Zo= 0.6. The importance of such wave
motions in the atmospherewould also imply a phasevelocity
distribution in the mesospherepotentially very different from
that in the lower atmosphere.However, wave motions that are
sufficientlylocalized horizontally are unlikely to undergo significantself-accelerationdue to rapid mean flow equilibration.
In an extensionof his earlier study, Lindzen [1984] attempted to provide an estimate of the level of wave activity and
thus the height above which saturation occurs for sheargeneratedand topographicallygeneratedgravity waves on a
global basis. Assuming a wavelength of ~ 1600 km and
average values for mean flow quantities, Lindzen concluded
that such waves should not break below ~70 and 60 km,
respectively.It should be pointed out, however, that this
analysis did not address other gravity wave sources,wave
superposition,transient effects,or smaller-scalegravity wave
motions,all of which may contributeto a loweringof the level
of saturation.Lindzen also addressedthe propagationof gravity waves with large horizontal wavelengthsin a nonzonal
mean flow and argued that such motions, by virtue of their
large horizontal propagation distances,could be horizontally
focusedor defoeusedby imposed variations of the intrinsic
phase velocity c- •. He suggested,therefore, that such motions could lead to substantial
variations
in the level of satu-
ration both zonally and meridionally. Similar focusingargumentsfor small-scalegravity wavespropagatingin an inertial
flow were advancedby Broutman[1982].
Some of the issuesdiscussedby Lindzen [1984] were examined in a quantitative manner by Schoeberland Strobel[1984].
These authors expandedthe one-dimensionalmodel of Schoeberl et al. [1983] to include molecular dissipation,a variable
stratification, and nonzonal mean winds. Assuming inertiogravity wave motions with horizontal wavelengths•h ~ 1000
km, Schocbcrl and Strobcl found no waves with horizontal
phasevelocitiesin the range
-10 < c < 30 m s-1
(43)
to reach the mesospherewith sufficientamplitude to saturate.
This was due to the strongradiative damping of suchmotions
with small intrinsic phase velocities,c - a. It should be noted,
however, that small-scale gravity waves with intrinsic frequenciesto >>f are not subjectedto strong radiative damping
(evenfor small c - a) and may reachthe mesosphere
relatively
unattenuated.Of more significanceis the result of Schoeberl
9O
120. - ' '
•o,'
ß
'
'
' ' ' - . •0
.,./..-•--.._. •
,....,.
"30
.... ..,',,
,
o
,
180
....
-f--:-'
--
ß
'
'
. .... o
,
,
'x/
210
,
' " '
ß
"•
"
.
ß
.
'
'
ß 330
,
,
.
,
27O
Fig. 19. Direction of average wave vector at 0.1 mbar due to
filteringof an isotropicdistributionof stationarygravity wavesabove
100 mbar [Dunkertonand Butchart,1984]. The vectorlength denotes
the fraction of wavessurvivingfiltering. The averagewave vector is
generallyalignedso as to opposethe averagewind in the filtering
layer.
290
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
•I
100
i •I
90
due to spatially or temporally varying mean flows may result
in regions of enhanced (lower) or nonexistentgravity wave
saturation [Fritts, 1982; Broutman,1982; Lindzen,1984; $choeberl and $trobel, 1984; Dunkertonand Butchart, 1984], and the
self-accelerationanticipatedby Grimshaw[1975], Coy [1983],
and Fritts and Dunkerton [1984] may produce mesospheric
phase velocities substantially different from those
characterizingthesemotions in the lower atmosphereif gravity wave motions are not sufficiently localized. Thus ample
motivationsexistfor studiesof the phenomenaassociatedwith
i
\
80
c:-zo-••
i
\\\
70 c=o
•,•\•-c:o
•
\\
60
/
local saturation.
50
40
20
I
-25
0
•1
25
I
•
I
50
75
I00
125
MEANZONALWIND(M •)
Fig. 20. Zonal wind profiles due to the saturation of gravity
waveswith c = 0 ms-:
and with c = 0 and -20 ms-:
[Holton,
1982]. The dashedcurve showsthe mean wind obtained with a Rayleigh drag model for comparison.(Reprintedwith permissionof the
AmericanMeteorologicalSociety.)
and Strobel that planetary wave motions can provide regions
within which gravity waveswith a given phasevelocity (topographically forced waves,for example) are favored to propagate unattenuated.The other phase of the planetary wave is
then a region in which such motions are preferentiallydissipated. The consequences
of this asymmetryin the mesosphere
include nonzonal drag and diffusionand the possibilityof in
situ planetary wave excitation.
In another study along theselines, Dunkertonand Butchart
[1984] addressedthe propagation and attenuation of both a
field of gravity waves with a specifiedorientation and an isotropic distribution of gravity waves at the 100-mbar level in
observedlarge-scalewind fields associatedwith stratospheric
warmings.Their findingsrevealedsubstantiallydifferentpropagation and attenuation dependingon the orientation of the
wave number at the 100-mbar level. It was found that quasistationary gravity waves with wave number vectors that do
not becomeorthogonalto the local mean flow may propagate
through regionsof easterlymean flow, implying a reduction
but not an elimination of suchwavesin a stratosphericwarming. In most casesthe wave packetsemergingat upper levels
had an averagewave number and thus a momentum flux so as
to oppose the average motion of the column through which
they had propagated. One example of the direction of the
average surviving wave vector resulting from a local analysis
of wave propagationis shownin Figure 19. If the wave field is
undergoingsaturation,the applied drag at the 0.1-mbar level
is seento be very nearly antiparallel to the local mean flow.
The effects of transience on gravity wave saturation have
been consideredin some detail in a number of fairly idealized
situations [Dunkerton, 1982a, b; Coy, 1983; Walterscheid,
1984; Dunkertonand Fritts, 1984; Fritts and Dunkerton,1984].
One peripherally related study that provided some insight
into the effects of local gravity wave saturation is that by
Walterscheidand Boucher[1984]. Though theseauthors consideredthe responseof the thermosphereto impulsivemomentum and thermal sourcesascribedto ion drag and Joule heating or particle precipitation,such a discussionwould seemto
apply equally well to the momentum flux convergenceand
thermal modificationsassociatedwith localized gravity wave
saturation. The conclusionsof this study were a rapid geostrophie adjustment due to a deep momentum source,with
inertio-gravity wave excitation favored for a shallow momentum source.In the caseof local heating, shallow sourcesproduced both geostrophic and inertial motions, while deep
sourcesproduced no significantresponse.These results suggest that gravity wave saturation near a critical level may
initiate inertio-gravity wave motions but that (deep)saturation
remote from a critical level may quickly attain geostrophic
equilibrium.
2.3. Nonlinear Saturation Theory
Becausethe gravity wave spectrum in the middle atmospherecan rarely, if ever, be consideredto be monochromatic,
a relatively complete treatment of the saturation problem
must take proper account of the interactionsamong gravity
wavesof large amplitude.The complexityof the generalproblem, however, has motivated most researchers to consider the
ramificationsof linear or quasi-linear saturation theories.
Nevertheless,there have been severalattemptsto considerthe
effects of various nonlinear
interactions
associated with or re-
100
7o
However, a related area, i.e., the effects of localized wave
-I•0 I - •0 I
0
packet saturation, appears to be equally significantbut has
ZONALFORCE(M S-I/DAY)
receivedvirtually no theoreticalattention to date. Suchstudies
are particularly important in light of the implicationsof some
Fig. 21. Zonal drag profilesfor the two gravity wave saturation
of the investigationsdiscussedpreviously.For example,the profilesand the Rayleighdrag profilein Figure 20 [Holton, 1982].
filtering and focusingor defoeusingof gravity wave motions (Reprintedwith permission
of theAmericanMeteorological
Society.)
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
suitingin gravity wave saturation.The first such attempt was
made by Weinstock[1976], and while a complete description
of this theory is beyond the scope of the present paper, the
significantassumptionsand conclusionsof this theory and its
subsequentextension [Weinstock, 1982] will be discussed.A
secondstudy by Lindzen and Forbes [1983], which addressed
the effect of one particular nonlinear interaction among gravity waves,will also be reviewedbriefly.
The generalcaseconsideredby Weinstock[1982] is that of
two-dimensional,steady state internal gravity waves that are
of sufficientlylarge amplitude that strong nonlinear interactions occur that exactly cancel wave growth due to upward
propagation.The mean flow was also assumedto be stationary and slowly varying in z. However, the nonlinear interactions were assumedto be sufficientlystrong that only turbulence resulted.
Resonant
triad
interactions
and the excitation
of other wave motions via nonlinearity were excluded from
this formulation. Thus the theory is nonlinear to the extent
that certain nonlinear terms are retained in the equations of
motion, but it is "linear" with respectto wave excitation and
wave-mean
flow interaction
291
o
80
+12
• 7o
•-
60-
ß.r 50
40
20I
i
630
i
720
I
810
i
900
TIME (days)
Fig. 23. Annual cycle of the zonal mean wind driven by transient,
stochasticgravity wave saturation [Dunkerton, 1982b]. The higher
easterlyjet core is causedby a weaker summer gravity wave spectrum. Units are meters per second.(Reprinted with permissionof the
American MeteorologicalSociety.)
because these effects are not in-
corporated.Under theseand other assumptions(including the
random phase approximation commonly used in turbulence
theory), Weinstock was able to show that the equations of
motion reduce to a nonlinear dispersionrelation identical to
the linear dispersion relation except for the inclusion of a
"nonlinear damping decrement" in the intrinsic wave frequency
ro- idw(k)
(44)
wave motions, dw is always positive. Clearly, this approximation applies only to wave motions with sufficientlylarge
amplitudesthat the effectsof nonlinearity are largely dissipative.
Using the expressionobtained for dw, Weinstock [1976,
1982] was able to calculatethe variation with height of each
component of the gravity wave spectrum in terms of the
characteristics
of the overall spectrum.This dependencepermitted calculation
of both the horizontal
and the vertical dif-
where dwis a function of the amplitudesand phasesof all the fusivitiesand the induced(but neglected)mean flow accelermotions that constitute the gravity wave spectrum. Because ations.A uniquefeatureof the nonlinearsaturationtheory is
nonlinear interactions were assumednot to produce other that becauseamplitudesare assumedto be limited by nonlinear interactions, convective and/or dynamical instabilities
within the wave field are not required to account for wave
saturation.Surprisingly,the broad spectrumresultsof Wein120
I
I
I
I
1
I
stock [1982] were in substantial agreement with the linear
monochromaticresultsof Lindzen [1981] for both the vertical
IiO
diffusivity and the form of the induced mean flow acceleration,
-AM = 245
(33) and (22), the principal difference being a somewhat
Co--I0
IOO
smaller vertical diffusion in the broad spectrumformulation.
The disagreementbetween forms of at and D in the linear
theoryand the narrow spectrumlimit of Weinstock'stheoryis
probably due to an inconsistencyof the broad spectrumassumptionswhen applied to monochromatic wave saturation.
E
Weinstockalso noted the particularly simplerelationshipbetween thesetwo quantities given by (34) and the fact that the
meanflow accelerationvariessubstantiallywith c - a.
Despite the complexityand the limitations of the nonlinear
saturation theory of Weinstock[1976, 1982], the theory has
severalappealing features.One significantprediction of the
,50
broad spectrumtheory was the considerabledisparityin the
40
.• P(c)
horizontal and vertical diffusivities associated with wave dissi-
pation. The disparityarisesfrom the largely horizontal parcel
displacementsdue to the (dissipating)gravity wave motion
WAVES
field. These resultswere shown by Weinstock[1976] to be
consistentwith the large horizontal diffusivitiesinferred from
-40-20
O
20
40
60
80
•OO •20
meteortrail observations[Greenhow,1959] while insuringsufficientlysmallverticaldiffusivitiesso as not to preventgravity
MEAN ZONALWIND(mg-I)
wave propagation.The smallervertical diffusivitiespredicted
Fig. 22. Zonal wind profilebeforeand after an averagesaturation by the nonlinear saturation theory are also more consistent
eventfor a phasespeedspectrumwith a meanof - 10 m s-1 [Dunkerton,1982b].As in Figure 20, easterlyphasespeedsproduceeasterly
zonal winds at upper levels.(Reprintedwith permissionof the American MeteorologicalSociety.)
with atmospheric observations.A seconduseful feature of the
nonlinear theory is the prediction of decelerationand diffusion
profilesthat vary smoothlywith height.Whereasone expects
292
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
MERN ZONRL WIND
DRY
350.
The first study to incorporatethe decelerationand diffusion
predictedby the linear gravity wave saturationtheory of Lindzen [1981] in a model of the middle atmospherewas that by
Holton [1982]. Holton used an initial-value, mid-latitude, •plane channel model with a specifiedradiative equilibrium
90
80
70
zonal
4O
3O
2O
90
75
60
45
30
1S
0
-15
-30
-45
-60
-75
-90
LRTITUDE
Fig. 24. Solsticezonal mean wind for a primitive equation model
of the middle atmosphere circulation including gravity wave saturation [Holton, 1983]. The grossfeaturesof the simulatedcirculation
are quite accurate despite the very crude gravity wave phase speed
spectrum.Units are metersper second.(Reprintedwith permissionof
the AmericanMeteorologicalSociety.)
a fairly sudden transition to saturation for steady monochromatic gravity waves, the nonlinear theory offers an explanation for the gradual onset of theseeffectsin the mesosphere
as well as the intermittent occurrenceof thin turbulent layers
observedin other regions of the middle atmosphere.It must
be remembered,however,that this theory is restrictedto only
certain typesof nonlinearityand doesnot includequasi-linear
or transient
effects. Therefore
it cannot be assumed to describe
generalsaturationproblemsor effects.
The study by Lindzen and Forbes [1983] addressedthe
possibilitythat a monochromaticgravity wave could be dissipated through a transferof energyto a pair of waveswith half
the vertical wavelengthvia the parametric subharmonicinstability mechanism identified by McCornas and Bretherton
[1977]. Assuming a rapid energy cascade, these authors
showedthis interaction not to be competitivewith convective
instability of the primary wave, suggestingthat this mechanism should not prevent the linear gravity wave saturation
addressedby Lindzen [1981]. Indeed, observationsindicate
that superadiabaticlapserates do accompanybroad spectrum
saturation [Philbrick et al., 1983]. The argumentspresented
by these authors, however, are severely restricted owing to
severalconsiderations.
The major limitation is that the parametric subharmonic instability examined by McComas and
Bretherton, which is primarily an interaction betweena largescale wave of large amplitude and two small-scalewaves of
small amplitude and half the frequency,may not representthe
most rapid energyexchangedue to resonanttriads. This study
also did not addressthe very challengingproblem of the effects of general wave-wave interactions among the various
componentsof an energeticspectrum of (saturated or unsaturated) waves.Indeed, this is one problem that has confronted
oceanographersfor some time [Garrett and Munk, 1972, 1975;
Thorpe, 1975; Miiller and Olbers,1975] and is now becoming
important in the atmosphereas well [Van Zandt, 1982]. More
will be said about these studies in section 5.
3.
GRAVITY
WAVE SATURATION EFFECTS
IN MIDDLE
ATMOSPHERE MODELS
Becauseof the relatively recent theoreticalattention and the
very scanty information currently available on representative
gravity wave parameters in the middle atmosphere,gravity
wave saturation effects have only been examined in a few
middle atmospheremodelsto date. Nevertheless,thesestudies
have served to illuminate the essentialrole of gravity wave
drag in balancingthe thermal and momentumbudgetsof the
middle atmosphere.
mean
wind
and a radiative
relaxation
time
that
de-
creasedfrom • 15 days at lower levels to • 1.5 days near the
mesopause.The expressions(22) and (33) were used to determine the decelerationand diffusion resulting from saturation
for a discretespectrumof zonally propagating gravity waves
havingphasevelocitiesof 0 and + 20 m s-•. To avoidunrealistic effects due to the sudden onset of deceleration
and diffu-
sion at the level of saturation, Holton assumedthese quantities to be continuous and to decay exponentially below the
level of saturation. It was found for the parametersusedthat a
zonal wavelengthof • 800 km provided the requisitedrag for
a steady state forcing configuration.Holton noted, however,
that (22) implies that gravity waves of shorter zonal wavelengthsneed only saturate sporadicallyto provide the same
zonal deceleration.
The winter solsticezonal velocity and zonal drag profiles
obtained by Holton [1982] for two choices of gravity wave
forcing are shown in Figures 20 and 21. The dot-dashedand
solid curvesrepresentthe steadystate saturation solutionsfor
thec=0ms-•
and thec=0and
-20ms-
• cases, respec-
tively. The mean wind and decelerationprofilesobtained with
the Rayleigh drag used by Holton and Wehrbein [1980] are
shown with dashedcurves for comparison.Relative to the
zonal velocity profile obtained in responseto the prespecified
Rayleigh drag, both of the zonal velocity profiles resulting
from parameterizedgravity wave saturation display substantially lower zonal wind maxima and stronger zonal velocity
shears above the jet core. These features resulted from the
substantiallylarger zonal drag exerted owing to gravity wave
saturation at lower levels of the flow (seeFigure 21). Because
the effect of gravity wave drag is to drive the mean flow
toward the phase speed of the gravity wave, that profile obtained with a wave forced at c-
-20
m s- • also exhibits a
reversalof the zonal wind at upper levels.This is one feature
of the observedmiddle atmospherecirculationthat is impossible to reproducewith a Rayleighdrag of the form Kt2.Because
the zonal wind is in approximate geostrophicbalance, the
large shearsabove the jet coresin the saturationmean wind
profiles also signify a much larger meridional temperature
gradient, and a correspondinglystronger meridional circulation, than that predicted with a Rayleigh drag, in better
agreementwith recentradiative heatingcalculationsand mean
meridional wind observations [Wehrbein and Leovy, 1982;
Apruzeseet al., 1982; Nastrorn et al., 1982]. Holton also succeededin reproducingthe grossfeaturesof the summer solsticecirculationand an annual cyclewith his very crudegravity wave sourcespectrum.
A very different approach to that of Holton [1982] was
adopted by Dunkerton[1982b]. Applying the conceptsdeveloped in previous studies [Dunkerton, 1981, 1982a], he succeeded in simulating the gross features of the mean zonal
circulation of the middle atmosphereby using a transient,
one-dimensional
wave-mean
flow
interaction
model.
The
principal approximation employed was that the mean flow
adjustmentdue to a saturatedwave packetcould be replaced
by that adjustmentdue to an unsaturatedwave packet encountering a critical level. Dunkerton also assumed a stochasticdistribution of phasevelocitiesfor gravity waveswith
equal incrementsof momentum and a radiative equilibrium
FRITTS: GRAVITY WAVE SATURATIONIN THE MIDDLE ATMOSPHERE
NAVE DRAG M/S/DAY
zonal velocity profile similar to that of Holton. The average
zonal velocityprofile obtainedfor winter solsticeand a mean
gravitywavephasespeedof c = -10 m s-• for onechoiceof
8O
wave packet wave action is shown in Figure 22. Here, the
profilesdenotedB and A representthe mean flow beforeand
after an average "saturation" event. The probability distribution for gravity wave phase speedsis shown in the lower
70
left. Like the results of Holton, those of Dunkerton indicate a
3O
rapid return of the mean flow toward the gravity wave phase
speedand a tendencyto producea reversalof the zonal velocity at upper levels.The larger velocity shearsabove the jet
2O
core in Dunkerton's
results relative to those of Holton
are a
consequence
of not smearingout the transition to saturation,
as done in Holton's steady state model. Also, illustrated in
Figure 23 is the annual cycle simulatedby Dunkerton. This
figure showsthe transition from winter westerliesto summer
easterliesin qualitative agreementwith observationsand the
influencein the lower winter thermosphereof gravity waves
with zonal phasespeedsc < 0.
The crude models of gravity wave saturation effectsin the
middle atmospherediscussedabove have shown these effects
to be relatively robust and insensitiveto specificfeatures of
the gravity wave source spectrum.Nevertheless,we should
anticipatethat the detailed responseof the middle atmosphere
to realistic gravity wave forcing will be considerablymore
complex.Until more is known about specificsaturation processesand characteristicgravity wave scalesand morphology,
however,it will be difficult to incorporatetheseeffectsin realis.ticlarge-scalemodels.
The first study to address the effects of gravity wave-
293
DAY
350.
-/40
'øf
LRT[TUDE
Fig. 26. The zonal drag neededto counteractthe Coriolis torque
acting on the mean meridionalcirculationat solstice[Holton, 1983].
(Reprintedwith permissionof the AmericanMeteorologicalSociety.)
and temperaturefieldsin Figures 24 and 25. Relative to observations, the simulation produced a westerly jet core in the
winter hemispherethat is ~ 7 km too low, a mean zonal wind
in the lower stratospherethat is too strong,and polar temperatures that are a little too cold. As was noted by Holton,
however,some of thesedeficienciescould be alleviatedby additional adjustmentof gravity wave parameters,but the large
unknownsconcerningthe atmospheregravity wave spectrum
and the possibledeficienciesof the radiative model suggest
that this would be inappropriate at the presenttime. Overall,
the model was very successfulat representingmost of the
grossfeaturesof the observedzonal mean circulation.In particular, the meridional temperature gradient in the upper
mesosphere,related to the vertical shear of the zonal mean
wind through the thermal wind relation, is reproducedvery
well, demonstratingthe departures from radiative equlibrium
induced deceleration and diffusion on the zonal mean circulation of the middle atmospherewas that conductedby Holton to be a direct result of the dynamical drag processesin the
middle atmosphere.
[1983]. In this study,Holton solvedthe primitive equationsin
The zonal drag provided by gravity wave saturation for this
sphericalcoordinatesto obtain the responseof the middle
flow
configurationis shown in Figure 26. The decelerationin
atmosphereto an idealized,nearly latitudinally independent
gravity wave sourcespectrum.The decelerationand diffusion the winter hemisphereboth beginslower and extendsover a
due to gravity wave saturation were parameterizedfollowing greater vertical distancethan that in the summer hemisphere.
gravitywaveswith c = 0 and -20 m s-• are
Lindzen [1981] with temporal and spatial smoothingnear the This is because
saturation level [Holton, 1982]. The radiative driving of the contributing, and saturating at different levels,in the winter
middle atmospherewas included using the schemeof Wehr- hemisphere.Only that gravity wave with c = + 20 m s-•
beinand Leovy [1982]. As in the work by Holton [1982], only avoidscritical levelsin the lower atmosphereand saturatesat
zonally propagating gravity waves with phase velocities of upper levelsin the summerhemisphere.The completeabsorpc = 0 and +_20 m s- • wereincluded.The amplitudesof these tion of all gravity wavesin this study,however,contrastswith
the findings of Schoeberland Strobel [1984], Dunkerton and
motions were adjusted to some degree to provide results in
Butchart [1984] and Fritts and Dunkerton[1984], which sugreasonableagreementwith observations.
Resultsof the zonally symmetricsimulation are shown near gestthat gravity wave drag may extend to substantiallygreatnorthern hemispherewinter solsticein the mean zonal wind er heightsin the atmosphere.The effectsof this drag distribution can be seen in Figure 24, where the decreaseof the
zonal mean wind with height above the westerlyjet core is
MEAN TEMPERATURE
DAY 350.
much more gradual than the correspondingdecreasein the
90
summerhemisphere.The mean meridional wind in this simu80
lation, with a maximum value of ~ 10 m s-• at ~75 km
7o •••---__..••
zoo'•
••-"•
220---•
directed toward the winter hemisphere,is also in much better
agreementwith observations[Nastrorn et al., 1982] than was
•o
•o-••
•o ••:
that obtained in previousmodels of the middle atmosphere
•o
••-•-••o
circulation that incorporated a Rayleigh drag to provide the
so
zonal momentum
2o
90
75
60
45
30
15
0
-15
-30
-45
-60
-75
-90
LATfTUBE
Fig. 25. Solstice temperature distribution in the middle atmospherecorrespondingto the zonal mean wind shown in Figure 24
[Holton, 1983]. Again, the grossfeaturesof the observedtemperature
field are reproducedwell. (Reprintedwith permissionof the American
MeteorologicalSociety.)
source.
Holton [1983] also performed several simulations of the
middle atmosphere circulation in the presenceof a forced
planetary wave. Despite the neglectof the modulation of the
gravity wave spectrum and levels of gravity wave saturation
by the planetary wave, he found that the planetary wave was
able to produce a significantmodification of the winter solsticezonal mean flow but was unableto providethe requisite
294
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE
....
x-•,•KIRUNA,
SWEDEN
"=•$•..
2200Z
10NOV'"'•
226Z
11NOV
,?
150.
175.
200.
225.
250,
275.
TEHPERATURE
(DEG
300.
325.
K)
ATMOSPHERE
butions to produce a reversal both of the solsticezonal mean
wind near 90-km altitude and of the mean meridional temperature gradient at the mesopause,in qualitative agreementwith
observationsand the resultsof Holton [1982, 1983] and Dunkerton [1982b]. In the context of the present paper, an eddy
diffusionschemeof the type usedby Matsuno may be thought
of as representingthe effectsof the saturation of a spectrumof
gravity waves with a broad distribution of phase speeds.A
similar study by Miyahara [1984] employedthe eddy diffusion
parameterizationof Matsuno to the zonal mean circulation of
the middle atmosphereat winter solstice.As in the study by
Holton [1983], these model resultsincluded a number of featuresin qualitativeagreementwith observations.
All of the models discussedpreviouslyhave addressedthe
effectsof gravity wave saturationprimarily in the mesosphere.
However, observations of the zonal mean flow and of turbu-
lencegenerationin the stratospheresuggestthat gravity wave
drag may play a role in the maintenanceof the zonal mean
wind in this region as well. This problem was addressedby
Tanaka and Yamanaka[1984] using the parameterizationof
gravity wave drag advancedby Lindzen [1981]. Assuminga
horizontal wavelengthof 200 km, a phase speeddistribution
27NOV '•,.•..•
2245Z.•
W
O.
28
NOV
..]•
-zt-ii'
47Z
329Z
centered about zero, and a momentum flux consistent with
that observed by Lilly and Kennedy [1973] occurring over
O.
10-40% of the globe,theseauthors found the induceddrag to
produce substantial easterly accelerationsof the zonal mean
flow throughout the stratospherein all seasons.These results
suggestthat it is important to obtain measurementsof momentum flux divergence and its variability throughout the
middle atmosphere.
't
175,
200.
225.
TEMPERATURE
250,
275,
300,
325,
(DEG K)
Fig. 27. High-resolution temperature profiles obtained (top)
beforeand (bottom) after a minor stratosphericwarming during the
WINE experiment [Philbrick et al., 1983]. Wavelike structuresand
superadiabaticlapse rates are seenthroughout the stratosphereand
mesosphere.
(Reprinted with permissionof PergamonPress.)
4.
OBSERVATIONS OF GRAVITY
WAVE
CHARACTERISTICS,
DISTRIBUTION,AND
SATURATION IN THE MIDDLE
ATMOSPHERE
decelerationin the absenceof gravity wave saturation.Finally,
under the conditions of a simulated stratosphericwarming,
During the last few decadesmany diversetechniqueshave
been developedto facilitate the study of gravity waves and
their effectsin the middle atmosphere.Early techniquesin-
Holton observedthe mesosphereto undergo a substantial
cluded measurements
cooling.This he attributed largely to the filtering of the gravity wave spectrumenteringthe mesosphereand a corresponding reductionin the zonal drag, due to the zonal wind reversal
(and formation of gravity wave critical levels)at lower levels,
as anticipatedby Lindzen [1981]. The reduceddrag, in turn,
permitted the mesosphereto relax toward a cooler temperature closer to radiative equilibrium, in line with observations
[Quiroz, 1969; Labitzke, 1972; Hirota and Barnett, 1977].
Mesosphericcooling may also occur as a result of planetary
wave-mean flow interaction near a critical line [Matsuno,
1971; Matsuno and Nakamura, 1979] and over greater heights
if the wave is transient [Schoeberland $trobel, 1980]. Nevertheless, the deceleration accompanying gravity wave saturation providesanother meansby which dynamicsmay influencethe thermal structureof the mesosphere.
A model similar to that by Holton [1982] was used by
Matsuno [1982] to examine the effectsof gravity wave drag
trails, chemical releases,and noctilucent cloud formations; the
of wave motions
observed
in meteor
inferenceof temperaturestructure using a seriesof rocketborne grenades;and determinationof wind and temperature
profiles from radar-tracked balloon and rocket payloads,
among others. More recently developed techniquesinclude
variousoptical methods;high-resolutionaircraft, balloon, and
rocket instrumentation;and ground-basedlidar and radar facilitiescapableof nearly continuousmeasurementswith good
spatial and temporal resolution. The new, high-resolution
measurementsystemsand thosecapable of continuousobservations over a broad height range, in particular, offer the
prospectof very exciting new data on the structureand dynamicsof the middle atmosphereover the next few years.For
additional discussionof the capabilitiesand limitations of the
various observationaltechniquesthe interested reader is referredto the workshopsummaryby Fritts et al. [1984b].
The purpose of this section is to review what has been
learned to date, using the various observational methods,
middle atmosphere. But unlike Holton, Matsuno assumed about gravity wave characteristics, their distribution and
gravity wave dissipationto occur through a prespecifiededdy variability, and the nature and effectsof gravity wave satudiffusion.This diffusion,like gravity wave saturation,has the ration in the middle atmosphere.We begin by describingthe
advantage that the mean flow is acceleratedtoward the phase spectrumof gravity wave motions observedin the middle atvelocityof the dissipatinggravitywave.Assumingan isotropic mosphere,together with representativewavelengths,periods,
gravity wave sourcespectrum,Matsuno found the differential and phasevelocities.Observationssuggestinglatitudinal, seadissipationresultingfrom the asymmetriczonal wind distri- sonal, and temporal variability of the gravity wave spectrum
and selective transmission
on the zonal mean motion
of the
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
.MAR•
25, Ig?2
295
80 km
•,o o
O0
GEOMAGNETIC
ms'•
MF..RIDIAN
-$0.0
1.0
VERTIC•
0.0
•-•
V•J.DClTY
-I.0
•,0.0
VELOCTTY AI.DNG THE
0.0
•l•Ol•lC
ms'e
EOU&TOR
-$0.0
1200
I:•00
1400
-I$OO
EST
Fig. 28. An example of high-frequencygravity wave motions at 80-km altitude observedwith the Jicamarcaradar
IRastoolandBowhill,1976a].(Reprintedwith permissionof PergamonPress.)
are then discussed.Finally, we review the evidenceof gravity
wave saturationin the middle atmosphere.
enceof wave motions with ;•z•' 2-20 km in the mesosphere
and lower thermosphere(Philbrick, private communication,
4.1. Observationsof Gravity Wave Characteristics
1983). Very small scale structures were also observed in the
stratosphere by Barat [1983] and Yamanaka and Tanaka
Early meteor trail observations by Liller and Whipple
[1954] and others revealedthe presencein the middle atmosphere of wavelike horizontal motions with vertical wavelengthsranging from • 2 to 15 km between 80- and 100-km
altitude (see Figure 1). Also apparent in these observations
was a tendencyfor the characteristicvertical wavelengthto
increasewith height. Similar wind structureswere observedin
the chemical release experiments performed by Kochanski
[1964], Blarnont and Barat [1967], and Woodrum and Justus
[1968]. Additional studiesby Theon et al. [1967] and others
using rocket grenade and pitot-tube sounding techniques
found similar vertical structuresto occur in the temperature
profilesbetween40- and 90-km altitude as well (seeFigure 2).
Becausethe temperatureswere inferred at 2- to 3-km intervals,
these measurements were not able to resolve structures with
[ 1984]usinghigh-resolution
balloontechniques.
The vertical wavelengthof a gravity wave motion is an
important quantity, because,with the assumptionsmade in
section2, it is seento yield a direct measureof the intrinsic
phasespeedof the wave motion provided that the mean stratificationN 2 is known,
Ic - al = N/m = N•z/2n
(45)
Thus a distribution of vertical wavelengthsin the middle atmosphereimplies some knowledgeof the range of gravity
wave phase speedsand of the potential for wave-mean flow
interactionin regionsof wind shear.The data shownin Figure
27 suggestthat gravity waveswith intrinsicphasespeedsrang-
ing from • 4 to 40 m s-x werepresentwhenthesedata were
collected.On the basisof the resultsof section2, however,we
;L=• 6 km. Recentlydevelopedrocket-bornetechniquesand might expectthosemotionswith larger verticalgroup velociground-basedradar and lidar systems,however, permit the ties and momentum fluxes (implying larger intrinsic phase
determinationof wind and temperatureprofileswith excellent speeds)to play a more significantrole in middle atmosphere
resolution.New rocket payloadsare now able to obtain wind
dynamics.
and temperatureprofileswith • 100-mresolution[Philbrick et
Studiesof temporal fluctuationsdue to gravity wave moal., 1983]. Radar and lidar facilitieswith resolutionsranging tionsin the middleatmospheredatebackto the early observafrom 50 m to 2 km offer the additional advantageof high tions of traveling ionosphericdisturbancesby Munro [1948,
temporalresolution,thus facilitatingstudiesof gravity wave 1950]. Subsequentstudiessuch as those by Weinsteinet al.
morphology and evolution in certain regionsof the middle [1966] and Cadet and Teitelbaurn[1979] usedsuccessive
balatmosphere.Observationsof gravity wave structure using loon launches to examine the temporal evolution of the
thesesystemshave confirmedthe dominantwavelengthsob- middle atmosphere.Not until the introduction of radar and
tained in those studies cited above [Manson et al., 1973; lidar measurementtechniques,however,did the potentialexist
ChaninandHauchecorne,
1981].The observations
by Philbrick for studiesof the entire spectrumof gravity wave motions,
et al. [1983], in particular, revealedthe presenceof fluctu- with intrinsicfrequenciesranging from the Brunt-V/iis/il/ifreations with vertical wavelengthsbetween •1 and 10 km
quency, N • 2•/(5 min), down to the inertial frequency,
throughoutthe winter mesosphere.
Correspondinglysmaller f <_2n/(12hours).The applicationof VHF radar techniques
to
structureswere observedin the stratosphere.An exampleof
atmosphericstudies was addressedby Balsley and Gage
thesedata is shownin Figure 27. Recentdata collectedin the
[1980] and Rb'ttoe•[1980]. Motions near the high-frequency
STATE experimentat 65øNduringsummerrevealedthe pres- end of the gravity wave spectrumhave been observedwith a
296
ERITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
different. With our assumptionsthesefrequenciesare related
by
• 23
co= f•-
k. u = k(c- if)
(47)
22
•
But it is the intrinsic frequencythat appearsin the approximate dispersionrelation
21
co2= k2N2/(k2 + m2)
F- 20
i
-3
i
-2
t
-1
I
I
I
J
0
1
2
S
WESTWARD
WIND
SHEAR
COMPONENT
(8-1) SOUTHWARD
WIND
SHEAR
COMPONENT
(8-1)
Fig. 29. An example of inertia-gravity wave motions in the
stratosphere[Barat, 1983]. Notice that the orthogonal wind shears
are nearlyin quadraturethroughoutthe heightrange.
number of techniques,including HF and VHF radars [Woodman and Guill•n, 1974; Rastogi and Bowhill, 1976a; Vincent
and Ball, 1977; Harper and Woodman, 1977; Fukao et al.,
1979; Czechowskyet al., 1979; Manson et al., 1981; Royrvik et
al., 1982; Vincent and Reid, 1983] and lidars [Chanin et al.,
1983] in the mesasphereand VHF and UHF radars in the
stratosphere[Riister et al., 1978; Watkins and Wand, 1981;
Sato and Woodman,1982b]. An exampleof the high-frequency
motions observedin the mesasphere( from Rastogi and Bowhill) is shown in Figure 28. The dominant periods range from
--•4 to 10 min, very near the high-frequencycutoff for internal
gravity waves. Low-frequency gravity wave motions in the
stratosphereand the mesaspherehave been observed using
balloon and radar techniques by Cadet and Teitelbaum
[1979], Frezal et al. [1981], Sato and Woodman [1982b],
Barat [1983], Balsley et al. [1983], and Yamanaka and
Tanaka [1984], among others. The zonal and meridional velocity shears observed by Barat during one strataspheric
soundingare shownin Figure 29. Notice that the two components appear to be in quadrature and indicate a downward
phaseprogression,suggestingan upward propagatinginertiagravity wave.
A number of authors have also computed the spectral
energy density of atmosphericmotions at various levels [e.g.,
Frezal et al., 1981; Vincentand Ball, 1981] and found a depen-
and which thus determinesthe gravity wave character. It is
clear from (47) that significantdifferencesbetweenthe intrinsic
and Doppler-shiftedfrequenciesmay occur if I•l is comparable
to [cl,and it is important that thesedifferencesnot confusethe
interpretations of data. Examples of Doppler-shifted frequenciesoutside the range of intrinsic frequenciesallowed for
vertically propagating gravity waves include the highfrequencywave packet near the beginningof the data record
in Figure 28 (for which fl > N), the low-frequency wave
motion observed by $ato and Woodman [1982b] for which
fl < f, and the smearingof the peak near co= N in the vertical
velocity spectra obtained by R6ttger [1981] due to nonzero
mean winds at some levels.
Other gravity wave parametersequally as important as vertical wavelength and period are horizontal wavelength and
horizontal phasevelocity.Without knowledgeof thesequantities it is difficult or impossibleto infer gravity wave effects.
Unfortunately, it is often difficult to make reasonableestimates of •.hand c from observationsat a singlelocation without detailed knowledge of the (nearly monochromatic)wave
field. Without such knowledge,three observationsseparated
by somereasonablefraction of the horizontal wavelengthare
neededto estimate•.hand c. There are, however,severalnatural phenomenathat facilitate the estimation of these parameters and observationof gravity wave morphology at certain
levelsof the middle atmosphereusing visual or optical techniques. Noctilucent clouds, which occur only at the highlatitude summer mesapause(• 85 km), are seen to exhibit a
wide range of wave motions whenever they are visible. The
photographic studiesby Witt r1962], Grishin [1967], Haurwitz and Fogle [1969], and others provided evidenceof horiPeriod
8d 4d ?d74hl2
h8h 4h
dence of the form
E(f•) ~ f•-•
(48)
(46)
fh
15m 3m
F-5/3
_
for an observedfrequencyf• and 1 < •c< 2. The zonal wind
speedpower spectralobtainedby Carter and Balsley[1982-]at
86 km for two summers are shown in Figure 30. The 1980
spectrum, in particular, exhibits significant peaks in energy
densitycorrespondingto the 8-, 12-, and 24-hour tidal components. At higher frequenciesthe spectra show fairly smooth
decayswith •c• 5/3. Spectra with very similar characteristics
were compiled by Frezal et al. at three widely separatedgeographical locations, using 10-15 days of meteor wind data
from each, suggestinga long-term homogeneityof the (assumed)gravity wavespectrumin the uppermesosphere.
Other
evidenceof a "universal"gravity wave spectrumin the atmosphereanalogousto that observedin the oceans[Garrett and
Munk, 1972, 1975-]was examinedby Van Zandt ['1982].
One very important point with respectto gravity wave frequencieswhich seemsto have been overlookedby a number
of researchersis that the observed(or Doppler-shifted)frequency f• and the intrinsic frequencyco(that frequencyseen
by an observermoving with the local mean flow) may be very
T
io
7 ,•
I
l0
G•
io
5 g,
- pokerFIotMST
Radar •
Summer Mesospheric
_
I0-G
86km
10-5
10-4
10
4 g
•.
Wind
Speed
Power
Spectra•
10-5 I
•
10-2
Frequency(Hz)
Fig. 30. The power spectraldensityof the zonal wind component
at 86 km during summerat Poker Flat, Alaska [Carter and Balsley,
1982].At frequencies
higherthan the tidal band the spectralslopeis
very nearly -5/3. (Reprinted with permission of the American
MeteorologicalSociety.)
FRITTS: GRAVITY
WAVE SATURATION IN THE MIDDLE
zontal wavelengthsranging from -.•10 to 75 km and horizon-
ATMOSPHERE
297
Wind
90-
tal phasespeedsof --•10-60 m s-2. The NLC observations
8
6
10
also indicated the presenceof Kelvin-Helmholtz instabilities
with horizontal wavelengths of --•3-10 km near the summer
mesopause.Similar observationsof gravity waves in the OH
emissivelayer occurringbetween80 and 90 km at lower latitudes were reported by Hersd et al. [1980]. These authors
observed horizontal wavelengths between --•30 and 100 km.
Another study by Armstrong [1982] revealed a nearly monochromaticwave motion with a wavelengthand a phasespeed
8
•
mi•
10
12•
ofapproximately
244km and72m s- 2,respectively.
Estimatesof horizontal phasevelocitiesand wavelengthsin
the mesospherehave also been made recently with the use of
two or three HF or VHF radar beams sampling locations
spaced20-35 km apart. The study by Vincent and Reid [1983]
obtained zonal wavelengthsof lessthan 200 km for the most
part, with an average value of --•70 km for periods of < 1
hour, at 35øS during the Austral winter. These authors suggested,however, that larger wavelengthsmay not have been
easilydetectedbecauseof their 35-kin beam spacing.The histogram of zonal wavelengthsinferred by Vincent and Reid is
shown in Figure 31. The zonal phase speedsinferred for these
i
J
i
i
F
M
i
i
A
M
I
J
i
J
i
A
I
S
6
i
i
N
D
MONTH
Temperature
90-
4
3
2
1.5
max
gravitywavemotionsrangedup to --•200 m s-2, with a mean
of -,•70 m s-2. Vincentand Reidalsonotedthat because
only
the zonal wavelengths and phase velocities were measured,
more realistic estimates for the mean values would be • 50 km
5
and --•50 m s-x, assuming
an isotropicspectrumof waves.In
a similar study using winter data at 65øN, Smith and Fritts
[1983] found evidenceof both low- and high-frequencygravity wave motions. Like Vincent and Reid, they found that
high-frequency gravity wave motions had horizontal wavelengths and phase velocities of --•20-80 km and • 10-50 m
4f
max
5
•
........
3.5_..-'"•.r•"'-.._
-•
min•,
L__
J
F
_x
M
I
I
I
I
I
I
I
A
M
J
J
A
S
O
I
I
N
D
MONTH
Fig. 32. Distribution of gravity wave activity in (top) wind and
s-2, respectively.
An anomalouslow-frequency
gravitywave (bottom) temperature as functions of latitude and season obtained
MRN data[Hirota,1984].Notethestrong
annualcycleat high
motion of large amplitude was also observedfor a period of 5 using
days.During this time the period was observedto evolvefrom
> 10 to -,•5 hours. The horizontal wavelength was likewise
observed to decrease with time. More
wave motions
in the remainder
will be said about these
of this section.
An additional feature of the observedgravity wave spectrum that has been noted by a number of authors is the tend-
0.3
0.2
0.1
and the semiannual
oscillation
at lower latitude.
Contours
are rms wind and temperature in meters per secondand degrees
Kelvin. (Reprinted with permissionof Terra Scientific Publishing
Company.)
ency for gravity wave motions to be polarized to somedegree
[Vincent and Stubbs,1977; Manson et al., 1981]. Vincent and
Stubbs suggestedthat this might be due to the directional
filtering of the gravity wave spectrumby the underlyingwind
field first addressedby Hines and Reddy [1967]. A preferred
orientation of gravity wave phase fronts is also suggestedby
the NLC observationsby Witt [1962], Grishin [1967], and
Haurwitz and Fogle [1969]; the OH emissionstudy by Hersd
et al. [1980]; and the recentradar studiesby Vincentand Reid
[1983] and Smithand Fritts [1983]. Suchobservationsappear
to be consistentwith the findingsof the recent study of selective transmissionof an initially isotropicgravity wave spectrum by Dunkertonand Butchart[1984-1,discussed
in the previous section.
I
I
o
latitudes
40
80
ZONAL
120
160
WAVELENGTH
200
240
> 240
/km
Fig. 31. Histogram of zonal wavelengthsinferred from dual-beam
HF radar studiesat Adelaide,Australia [Vincent and Reid, 1983]. The
mean zonal wavelengthwas • 70 km. (Reprinted with permissionof
the AmericanMeteorologicalSociety.)
A final characteristicof gravity wave propagationnoted by
severalauthorsis the apparentchangein the observedgravity
wave frequencywith height [Rastogi and Bowhill, 1976a; Stening et al., 1978; Mansonet al., 1979]. Sucha changecannot be
explainedin termsof linear theory for a monochromaticgravity wave, and it is not clear that the spectralresolution in
theseobservations
wassufficientto distinguishbetweenclosely
spacedfrequencies.Nevertheless,suchfrequencychangesare
predictedby the quasi-lineartheory for monochromaticgravity waves[Coy, 1983; Fritts and Dunkerton,1984-1,suggesting
298
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
I
I
20
11
OCT
I
21
22
1979
TIME
I
23
(AST)
I
I
0
1
12
OCT
1979
Fig. 33. Time-heightcrosssectionof turbulenceintensityin the lower stratosphereat Arecibo,Puerto Rico [Sato and
Woodman,1982a]. In addition to the nearly stationarylayersbetween~ 16 and 20 km, there are regionswith downward
phaseprogression.
(Reprintedwith permissionof the AmericanMeteorologicalSociety.)
that
these
observations
cannot
be dismissed
without
ad-
ditional study.
4.2. Gravity Wave Variability
Perhaps the most striking feature of the gravity wave spectrum in the middle atmosphereis its enormousvariability. The
variable nature of the spectrum may be attributed to a
number of factors, including, among others, intermittent or
episodic sourcesand propagation through and interaction
with a complex environment. Given the significant role of
gravity wave saturation processesin middle atmosphericdynamics,this variability also constitutesone of the major hurdles for those attempting to produce realistic models of the
generalcirculationof the middle atmosphere.
One of the first indications of the substantial latitudinal
and
seasonalvariability of the gravity wave spectrumin the middle
atmospherewas the study by Theon et al. [1967], which used
rocket-borne grenadesto infer the temperature structure due
to gravity wave motions. These observations,made at Wallops Island (38øN), Churchill (59øN), and Barrow (71øN), suggestedthat gravity wave fluctuationsare present at all latitudesduring winter, with the greatestlevel of activity at high
latitudes.During summer,in contrast,some gravity wave ac-
tivity was observedat lower latitudes,but virtually no wavelike temperature fluctuations were found at high latitudes.
These differencescannot be explained solely in terms of the
variation of the temperaturefluctuationsof the wave motions
with the Brunt-Viiisiiliifrequency,5T ~ N 3/•, and thusmust
indicateseasonaland latitudinal changesin the level of gravity
wave activity [Balsleyet al., 1983]. Additional evidenceof the
strong seasonalvariability of gravity wave energy was subsequently obtainedat a number of locationsby usingHF, VHF,
and meteor radar systems[Manson et al., 1975, 1979, 1981;
Frezal et al., 1981; Balsley et al., 1983]. In each of thesestudies, gravity wave activity in the mesospherewas found to be
significantlygreaterin winter than in summer.
In a more comprehensivestudy, Hirota [1984] examined
the gravity wave fluctuationspresentin the wind and temperature fields obtained using all active northern hemisphere
meteorologicalrocket network (MRN) sitesover many years.
Assuminga dominant vertical wavelength•:-• 10 km and
using available data between 20 and 65 km, he observed a
notable annual cyclein the rms wave amplitudesat high latitudeswith the maximum activity during winter, in good agreement with the observations cited above. These results are illus-
trated in Figure 32. Another interestingfeature of thesedata is
the semiannual cycle at lower latitudes with the activity
maxima occurring near the equinoxes.Clearly, this complex
mean seasonal and latitudinal distribution of gravity wave
activity is the product of a number of atmosphericprocesses
that must be understoodin greater detail before their middle
atmosphereeffectscan be anticipated.
In addition to the seasonaland latitudinal variability discussedabove,there is evidenceof substantialtemporal variability of the gravity wave spectrumin the middle atmosphere.
Gravity wave amplitudes,wavelengths,phasevelocities,orientation, and spectralwidth have all been observedto change
significantlyon time scalesranging from a wave period to
severaldays [Witt, 1962; Grishin,1967; Haurwitz and Fogle,
1969; Carter and Balsley, 1982; Vincent and Reid, 1983; Balsley et al., 1983; Hirota, 1984; Smith and Fritts, 1983]. Two
examplesof such variability are the rocket and radar data
shownin Figures27 and 28. In fact, it is often challengingto
find, in data with high spatial or temporal resolution,wave
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
motions
that retain
their character
North
for more than a few wave
periods. Becausewave motions with different characteristics
may be superposed,it can be difficult to identify the dominant
gravity wave motion unambiguously.This could prove to be
troublesomebecauseit is not clear that an analysisbasedonly
on those wave motions that are sufficientlymonochromaticto
be identified will be entirely representative.Nevertheless,it is
important to learn the characteristicsof those gravity wave
motions that appear to contributemost to middle atmosphere
processes
in order that their effectscan be understood.
299
- South
IiO
\
ioo
ß
9o
E
80
J o n u o r y 1976
i
ßo
i i illill
i
! i i l,,,l
1 ./'
I10
i'•,•'•'• '•
,
i
i i Illi!
I
I
,,lllll
t
ioo
4.3. Evidenceof Gravity Wave Saturation
Despite the recent recognitionof the important role that
gravity wave saturationplays in middle atmospheredynamics,
evidenceof gravity wave saturation has been available for 2
decades.Early indicationsof wave dissipationinclude,among
others,numerousobservationsof high levels of turbulent diffusion, measurementsof superadiabatic lapse rates in the
mesosphereand lower thermosphere, and observations of
Kelvin-Helmholtz billows near the summer mesopause.The
turbulent diffusionobservationswere suggestiveof horizontal
and vertical mixing • 100 times strongerthan that expected
for moleculardiffusion,indicativeof a largeenergydissipation
rate [Blamont and de Jager, 1961; Zimmermanand Champion,
1963; Blamontand Barat, 1967]. Rocket grenadeand ion trap
temperaturemeasurementsby Theon et al. [1967] and Knudsenand Sharp [1965] provided evidenceof convectivelyunstable layersdue to large-amplitudegravity wave motions.And
the frequent occurrencesof Kelvin-Helmholtz instabilities at
the high-latitude summer mesopausesuggestedby the NLC
observationsof Witt [1962], Grishin[1967], and Haurwitz and
Fogle [1969] imply a dynamical instability of the mean and
wave motions in that region.
More recent observationsutilizing a number of techniques
have providedadditional evidenceof gravity wave saturation
throughout the middle atmosphere.Estimatesof eddy diffusion due to turbulent layers in the lower stratospherewere
obtained using aircraft, rocket vapor trail, and balloon
measurementtechniquesby Lilly et al. [1974], Rosenbergand
Dewan[1975], and Cadet [1977]. Theseestimatesrangedfrom
,•
9o
• 12 HR •.
a Plonelory Woves
8o
6 Grovify Woves
,.• (kg•'s •)
Fig. 34. Energy density of planetary waves, tides, and higherfrequencygravity wavesduring winter at Saskatoon(52øN) [Manson
et al., 1979]. Notice the gravity waves' dominanceand the energy
decaywith height.(Reprintedwith permissionof PergamonPress.)
tion of an inertio-gravitywave with an amplitudeand vertical
wavelengthof u'• 5 m s-• and •l• • 1.5 km in the same
height interval [Sato and Woodman,1982b]. Assumingthat
the intrinsicfrequencyis sufficientlylarge that (48) is valid, the
observedwave structure correspondsto an intrinsic phase
speedof
c - t2,-, 5 m s-•
(49)
suggestingthat the wave motion may be saturating via dynamical or convective instabilities, as discussedin section 2.
magnitudelarger than that obtainedby Lilly et al. It is possible, however,that thesedifferencesmay be attributed to different geographicallocationsand the inability of a radar at one
locationto infer a spatiallyaveragededdydiffusion.
The distribution and intensity of turbulenceobservedby
Sato and Woodman[1982a] for a 7-hour period is shown in
Figure 33. This figure suggeststhat turbulencein the stratospheremay occur in thin, isolatedlayers.In someregionsof
the figure there is evidenceof the downward motion of turbulent layers,suggestiveof upward propagatinggravity waves
(assumingIcl > Il). It is also interestingto note that the most
Additionalevidenceof this is providedby the apparentlack of
amplitudegrowthof the wavemotion with height,the stationary nature of both the wave motion and the turbulent layers
in that region,and the spacingbetweenthe major peaksin the
echo power profile of • 1.5 km. Thus the most intensestratospheric turbulence observedby Sato and Woodman[1982a]
may be linked directly to the saturationof an inertio-gravity
wave motion. Like Sato and Woodman,Barat [1982] found
the regionsof maximum turbulenceto correspondto the regionsof maximumsheardue to the wave motion, supporting
the contentionthat low-frequencymotionsmay saturatevia a
dynamical (or sheer)instability rather than via a convective
instability [Balsley et al., 1983].
As in the stratosphere,recent radar observationsof waves
and turbulencehave providedadditionalevidenceof gravity
wave saturation throughout the mesosphere.A number of
studieshave revealedthe presenceof turbulent patchesor
layers [Rastogi and Bowhill, 1976b; Harper and Woodman,
1977; R6ttger et al., 1979]. Czechowskyet al. [1979] and
Royrvik et al. [1982] also observeda tendencytoward downward motion of someof the turbulent layers.On the basisof
the observedturbulencestructure,Harper and Woodman suggestedthat the turbulencemight originatefrom the dissipation
of high-frequencygravity wave motions. Other radar studies
of the gravity wave spectrumin the mesosphereprovided
somesupportfor this conjecture.The energyin the gravity
wave spectrumwas observedto decay rapidly with height
throughout the mesosphereby Manson et al. [1974] and
Manson et al. [1975]. In the latter study, a correlation was
intense turbulence
observed between the level at which the zonal mean wind
averagevaluesof •0.01-0.2 m2 s-• to local valueswithin
turbulentlayersof • 1-10 m2 s-•, depending
on the measurement techniqueand the environmentin which thesemeasurements were made.
Atmosphericradars have also been used to examine the
structureand intensityof turbulent layersin the stratosphere
and mesosphere[Cunnold, 1975; Gage et al., 1980; Watkins
and Wand, 1981; Sato and Woodman, 1982a; Wand et al.,
1983]. The studiesby Gage et al. and Sato and Woodman
provided estimatesof an averageturbulent diffusivity in the
lowerstratosphere
of •0.2 m2 s-•, approximately
an orderof
was observed between
• 15- and 20-km
altitude during the time immediatelyfollowing the observa- passed through zero and the level at which the maximum
300
FRITTS.' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
16-
the degreeof polarizationof the observedwave field,predicted
an acceleration
of if, • 10 m s-x d-x. Thusboth of the prin12
cipal effectsof gravity wave saturation,mean flow acceleration
and induced diffusion, had been anticipated and estimated
prior to the recognition of their fundamental role in middle
atmospheredynamics.
The high-resolution temperature measurements recently
conductedby Philbrick et al. [1983] provide additional evidenceof gravity wave saturationthroughout the winter stratosphereand mesosphere.Those data, a portion of which is
shown in Figure 27, indicate that superadiabatic(and convectively unstable)regionsare commonly associatedwith gravity
wave motions with vertical wavelengthsranging from • 1 to
10 km. In these data the dominant vertical wavelengthin the
stratosphereappears to be • 1-2 km, while that in the mesosphere is •3-10 km. Superadiabatic lapse rates were also
observedto be associatedwith vertical wavelengthsof ,•z •
2-4 km near the high-latitude summer mesopause(Philbrick,
personal communication, 1983). Consistent with the observations of Theon et al. [1967], there is little evidenceof multiple,
large-scale superadiabatic lapse rates in these data. In line
8
0
with
Fig. 35. Estimates of vertical flux of zonal momentum with 4-km
resolutionat three heightsaveragedin 6-hour blocksobtainedwith
the dual-beamtechnique[Vincent and Reid, 1983]. The variability of
the flux is remarkable. (Reprinted with permissionof the American
Meteoerological Society.)
the discussion
in section
2 the smaller
vertical
wave-
lengths in the stratospheresuggestsmaller intrinsic phase
speedsand smaller mean flow accelerationsthan those anticipated in the mesosphere.However, it is clear from these data
that a broad spectrumof wavesparticipatesin saturationprocessesin the middle atmosphere.
4.4. Measurementsand Analysesof Gravity
Wave Saturation
Very few observationalstudieshave specificallyaddressed
the saturationof gravity wavesin the middle atmosphereto
gravity wave amplitudeswere observed.These authors offered date. One obviousreasonis that the linear saturationtheory
an explanation for this observation in terms of the critical- that provides the framework within which the effects'of satulevel absorption of the various componentsof the gravity ration can be calculatedwas only recentlyavailablein a genwave spectrum, anticipating some of the features of the cur- eral form [Lindzen,1981]. A more significantreason,perhaps,
rent saturation theory. Other studiesby Vincent and Stubbs is the difficultyin determininggravity wave parametersin the
[1977] and Manson et al. [1979, 1981] provided estimatesof mesospherein an unambiguousmanner. Nevertheless,several
the turbulent diffusionbasedon the rate of decaywith height efforts have succeededin providing estimatesof gravity wave
of gravity wave energy.The decay of energywith height ob- parametersand effectsthat may be usefulboth in the designof
tained by Manson et al. [1979] for variouscomponentsof the additional observationalstudiesand as indicationsof imporfrequencyspectrumis shown in Figure 34. Typical valuesof tant processesrequiring theoreticalattention.
the energydissipationrate and the correspondingverticaldifThe first attempt to infer the decelerationdue to gravity
fusioncoefficientinferredfrom the observedenergydecay in wavesaturationand the corresponding
gravitywaveparamethe 80- to 100-km altitude rangeare
ters in a systematicmanner was that by Vincent and Reid
• • 0.01- 0.1 m2 s-3
(50) [1983] using a partial-reflectionradar in Adelaide, Australia.
These authors circumvented the need to measure individual
and
D • 200- 600 m2 s-x
(51)
which are consistentwith estimatesobtainedusingother tech-
niques[Hines,1965;Reeset al.,1972].It shouldbenotedthat
gravity wave parametersby devisinga systemby which the
momentumflux, and hencethe momentumflux divergence,
due to an arbitrary spectrumof gravity wavescould be measureddirectly.This systemutilizestwo Doppler radar beams
inclinedat equal and oppositeanglesfrom the vertical.The
vertical flux of horizontal momentum in the plane of the
beams is then given in terms of the differenceof the mean
thesevalueswere obtained using spectralexpressionsderived
by Zimmermanand Murphy [1977] becausecorresponding
estimates assumingmonochromaticwave motions proved excessivelyhigh [Manson et al., 1979]. Using an expressionobtained by Hines [1972], Manson et al. [1975] and Vincent and
Stubbsalsoestimatedthe mean flow accelerations
inducedby
the dissipation of a polarized gravity wave spectrum.Assumingall of the observedwave energyto be associatedwith a
single,coherentwave motion, Manson et al. [1975] obtained
periods.
Using the technique described above, Vincent and Reid
[1983] obtained estimatesof the vertical flux of zonal momen-
estimates of the mean flow acceleration in the 90- to 100-km
tum with 4-km vertical resolution between 78- and 98-km
altituderegionas high as tit • 200 m s-x d-x. A secondesti-
altitudefor 3 consecutive
daysin May 1981(Australwinter)at
mate by Vincent and Stubbs,which attemptedto accountfor
35øS.These data, averaged in 6-hour blocks, are shown at
square Doppler velocitiesobservedin the two beams.No decompositionof the velocity field into horizontal and vertical
componentsis required.The only requirementis that the data
intervalusedfor the analysisbe longerthan the gravitywave
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
three levels in Figure 35. Perhaps as a result of the nearly
u •w• / m2 s-2
continuous occurrence of wave motions with T < 2 hours, the
100
6-hour-averagedmomentum fluxes experiencedconsiderable
temporal fluctuationsin magnitudeand sign. Fluctuations of
particularly large amplitude were observed at the higher
levels. The vertical flux of zonal momentum normalized by
densityand the zonal flow accelerationinferred from the flux
divergenceaveraged over the 3-day observation period are
shown in Figure 36. We see from this figure that both the
densitydecreasewith height and a changein the amplitude or
character of the gravity wave spectrum have contributed to
the induced mean flow acceleration,as anticipated in section
2. The zonal accelerationis seen to vary between -12 and
-22
m s- • d- • between 80- and 94-km altitude. Thus the
effect at all levels was a deceleration
at~-17to
-25ms- •d- •
(52)
-1.0
-0.5
i
I
I
I
-20
-10
ß
0
F / ms-1doy-1
of the winter mean west-
erly flow. It was also noted by Vincent and Reid that the
actual zonal decelerationwas probably somewhatlarger,
301
Fig. 36. Vertical flux of zonal momentum and induced zonal drag
obtained with the dual-beam techniqueaveraged over 3 days during
the Austral winter rVincent and Reid, 1983]. (Reprinted with permissionof the American Meteorological Society.)
becauseof the broad radar beams and the tendencyfor enhanced near-vertical
reflections.
The magnitude of the zonal decelerationobtained by Vincentand Reid [1983] is somewhatsmaller than the mean value
thought to be required to counteract the effect of the Coriolis
torque on the mean meridional circulation. There are, however, severalpossibleexplanationsfor the shortfall. One is the
considerablevariability of the gravity wave spectrumevident
in the data shown in Figure 35 and noted by a number of
other observersas well. Large-scale,low-frequencymotions
are thought by someto provide the dominant momentumflux
and decelerationin the mesosphere[Lindzen, 1981, 1984]. It
can be argued, nevertheless,that low-frequencymotions with
large amplitudes (and large fluxes) are relatively infrequent
[Fritts et al., 1984a] and should not dominate the vertical flux
of momentum for observedgravity wave spectra (this argument is presentednear the end of this section).The observations of high- and low-frequencygravity wave motions by
Vincent and Reid support the latter view. This technique,
however,is unable to determine accurately horizontal wavelengthslonger than •n ~ 200 km. A more likely possibilityis
that the directionalfilteringdue to lower level windsproduced
a spectrum of saturating gravity waves not polarized in the
zonal direction [Smith and Fritts, 1983; Dunkerton and Butchart, 1984]. In such a case the inferred zonal deceleration
would be somefraction of the applied drag. This uncertainty
suggeststhat future flux and deceleration measurements
•z ~ 12 km. From the approximate gravity wave dispersion
relation we find a characteristicintrinsicphasevelocityof
c- a = N/m ~ 35 m s- •
(53)
in close agreementwith the observedrms velocity amplitude
and the predictions of linear and nonlinear saturation
theories.
Additional observationsand estimatesof saturating gravity
wave parameters were made using the MST (mesosphere,
stratosphere,troposphere)radar at Poker Flat, Alaska (65øN).
This facility has operated nearly continuously for several
years,collectingdata in the troposphere,lower stratosphere,
upper mesosphere,and lower thermosphere.A remarkablefeature of the data set is the dramatic seasonaldependenceof the
mesosphericand lower thermosphericecho region [Ecklund
and Balsley, 1981]. During winter, radar reflections are obtained between ~ 55- and 80-km altitude. During summer,on
the other hand, echoes are found between ~ 78- and 100-km
altitude. These observations,together with the markedly different temperature structure found at high latitudes during
winter and summer by Theon et al. [1967-1,suggestedsomewhat differentsaturationmechanismsoperatingduring winter
and summer [Balsley et al., 1983].
The winter data, includingthe temperaturedata of Theon et
al. [1967], are generallysuggestiveof gravity wave saturation
via the convectiveinstabilityof the wave field. As in the study
should be made in both zonal and meridional directions. Finby Vincent [1984], a dominant vertical wavelengthof ~ 10-15
ally, measurementsmade using this techniquedo not include km was observed,and the wind variability (rms velocity amthe contribution to the momentum flux of stationary (c = 0) plitude) was found to be approximatelyconstantwith height
gravity waves,whichmay be a significantfraction of the total.
[Balsleyet al., 1983]. The wind variabilityand spectralwidth
In a subsequentstudy, Vincent [1984] provided further evi- obtained for six 10-hour periods between October 14, 1980,
dence of gravity wave saturation at Adelaide (35øS) and and January 5, 1981, are shown in Figure 37. The wind variaTownsville(19øS),Australia. He performeda rotary spectral bility is that of the radial velocity.The rms amplitude of the
analysisto examinethe fraction of low-frequencywave energy horizontalwindis 4 timeslarger,~ 30-35 m s-•, in agreement
that was upgoing and downgoingand found a lower limit of with the value obtained by Vincent. Thus these observations
65% to be associatedwith upward propagatinggravity waves. also appear to be consistentwith the predictionsof saturation
Vincent inferred a vertical flux of energyat the mesopauseof theory. Another indication of gravity wave saturation in the
~ 10-2 W m-2 due predominantly
to high-frequency
(T < 1 winter mesosphsere
is the tendencyfor the mean zonal wind
hour) wave motions and an eddy diffusioncoefficientD ~ 300 to decrease
from a maximumwesterlyvalueof ~ 35 m s- • at
m2 s- • basedon the diffusionexpression
derivedby Wein- 60 km to valuesnear zero at ~ 80-km altitude [Balsley et al.,
stock [1982]. Finally, Vincent obtained strong evidence of 1983]. In the absenceof gravity wave drag, the mean zonal
gravity wave saturation,with an observedrms velocityampli- wind would continueto increasewith height.
tudeof ~ 30-35 m s- • anda dominantverticalwavelength
of
The MST data have also been usedin the analysisof indi-
302
FRITTS' GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
averageestimatedintrinsicphasespeed,we find that
v'
Averagevaluesof
spectral w•dth and
w•ndvariability for s•x _
ten- hour periods
15
« Ic- 01
(57)
This wave amplitude is substantiallylessthan that required
for the formation of dynamicalor convectiveinstabilitiesin a
monochromatic wave, suggestingthat saturation is broad
spectrumwith both low- and high-frequencygravity waves
contributing. Those higher-frequencycomponentsidentified
between 14 Oct 1980
and 5 Jan 1981
.• 70
._
by Smith and Fritts [1983] were found to have horizontal
65
wavelengths
and phasespeedsof ~ 50 km and •<50 m s-x.
--'--
--•--
Likewise,the mean flow accelerationinducedby the 10-hour
waveis, from (18),
Spectral Width _
t < 3m
Wind Variability
3m<t<l h
t7t • -25 m s- x d- •
550z 4 6 8 •o •z •4
(58)
Width (m/s)
The magnitudeof this accelerationis substantiallylessthan
the value that is thought to be necessaryto counteractthe
Coriolis torque acting on the mean meridional circulation
[Lindzen, 1981; Holton, 1982]. This suggeststhat the mean
flow accelerations
in the middleatmosphereare accomplished
vidual wave motions to examine the nature of the saturation
more efficientlyby those higher-frequencymotions that are
mechanism.One such wave event observedby Balsleyet al. presentalmost continuously.
Becausethe implicationsof mesospheric
drag and enhanced
[1983] is shownwith consecutive1-hour averagesof the horizontal velocitydata in Figure 38. It shouldbe stressed,how- diffusiondue to low- and high-frequencygravity wave moever, that suchnearly monochromatic,large-amplitude,low- tions may be very different,it is usefulto estimatethe relative
frequencywave motions are not commonlyobserved.In most momentumflux due to each.Following a suggestionby I. M.
casesthe vertical and oblique radial velocitiesare dominated Reid and R. A. Vincent (private communication,1983), we
by motionsat substantiallyhigherfrequencies.Nevertheless,it assumea spectralenergydensity for horizontal motions of the
is instructiveto considerthe implicationsof such a gravity form
Fig. 37. The spectral width and wind variabilty obtained at
Poker Flat, Alaska, during winter [Balsley et al., 1983]. The nearly
constantvaluesimply gravity wave saturation.
wave. The characteristics of this wave motion were determined
for 3 successivedays by Fritts et al. [1984a] and Smith and
Fritts [1983].
The inertio-gravity wave was initially observed to have a
vertical wavelength and a period of • 14.5 km and • 10
hours, respectively.A least squaresfit to the horizontal and
vertical velocity data revealed perturbation velocity amplitudes of
(u', v', w') = (14.2,19.6,0.18)m s-x
(54)
that were approximatelyconstantwith height.The resulting
phasedata illustratedin Figure 39 showedthe zonal and meridional perturbationvelocitiesto be very nearly in quadrature
with the northward and verticalvelocitiesalmostentirelyout
of phase,implying propagationtoward the south.An estimate
of the horizontal wavelengthin the directionof propagation
was obtainedfrom the ratio of horizontaland verticalperturbation velocitiesin this direction,
;•n• (v'/w'))•z• 1590 km
E(c0)~ ro-K
(59)
as observedby Carter and Balsley[1982]. Then relating the
verticalvelocityto the horizontalvelocityby usingthe dispersion relation,
w' • (co/N)u'
(60)
the incrementof momentumflux normalizedby densityat a
particularfrequencymay be written
tS(U'W')
"-'((.o/m)u
'2 ~ (.ol-K/m
(61)
Selectinga period T as the divisionbetweenlow- and highfrequencymotions,we integrate(61) to obtain the two contributionsto the total flux. The ratio of theselow- and highfrequencymomentum fluxesis
R=
(U,W,)t
' (2/r/T)2-• _f2-•
•
(u'w')u N 2-• - (2•r/T)2-•
(62)
(55)
The estimatedphasevelocity,
c = 2nIT • 44 m s-x
(56)
was observed to be • 30% smaller than that inferred from the
Poker Fiat, Alaska
11 October1981
HorizontalWindProfiles(one-hour
average
values)
Northward
vertical wave structure and the average mean meridional
motion,t•• -12 m s-x. Thismaybeduein partto thehighly
variablemean motion with height.Similar analyseswere per-•?
Eastward
formed to determinethe wave structurefor the following 2
•- 80
days as well. The most notablefeaturesof the inertiø-gravity
wave evolution were the substantialreductions(by about 2)
observedin the wave period, the horizontal wavelength,and
6
the perturbation velocity in the direction of wave motion.
Throughoutthe evolutionthe wave was found to propagate
Fig. 38. An anomalous,low-frequencygravity wave motion that
largely toward the south.
was observedon October 11, 1981 [Balsleyet al., 1983]. Individual
Comparing the amplitude of the wave motion with the profilesare 1-houraveragesof velocitydata.
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
Thus the ratio is determinedonceN,f, T, and tcare specified.
For tcsufficientlylessthan 2 and
f<< 2•:/T <<N
Poker Flat, Alaska
303
29 June 1981
HorizontalWindProfiles (one-hour
averagevalues)
Northward
(63)
(62) may be approximated as
R ~ (2•/NT) 2-"
(64)
Eastward
In general,however,•cis large enoughthat this is not a good
approximation[Frezal et al., 1981; Vincent and Ball, 1981;
Carter and Balsley, 1982; Vincent, 1984]. Assuming a
Brunt-Viiis•l/ifrequencyN = 0.02 s-x, an inertial frequency
Fig. 40. An example of the 1-hour-averagedsummer wind field
f= 2n/(13 hours), an energy spectrumwith •c= 5/3 [Carter
and Balsley, 1982], and a period T = 2 hours, corresponding near the mesopauseat Poker Flat, Alaska [Balsley et al., 1983].
Clearly, a number of wave motionsare presentin thesedata.
to a horizontal wavelengthof •h ~ 200 km [Vincent and Reid,
1983], we obtain from (62)
R ~ 0.25
(65) evidenceof a dynamical instability of some portions of the
wave field near the summer mesopause.These observations,
Valuesof N - 0.01 s-x andf- 2n/(24hours)yield a ratio of togetherwith the lack of large-scalesuperadiabaticlapserates
R ~ 0.45. This analysis implies that low-frequency motions in the summer temperaturedata collectedat high latitudes by
(with T > 2 hours) contributelessthan 1/3 of the total mo- Theon et al. [1967] and the detection of a strong daily periodmentum flux in the middle atmosphere,in accordancewith the icity in the intensity of the summermesopauseecho,led Balsconclusionsof Vincent and Reid [1983] and Smith and Fritts ley et al. to conclude that a significant portion of the dissi[1983].
pation of gravity wave and tidal motions near the summer
Like the winter mesosphericdata, those collectednear the mesopauseis accomplishedby the shear instability of lowsummer mesospauseexhibit considerablewave activity with frequencymotions rather than by the convectiveinstability
typical vertical wavelengthsof ~ 5-20+ km. An example of addressedby Lindzen [1981]. The occurrenceof shear instathesedata is illustrated in Figure 40, which shows 16 consecu- bility, of course,does not preclude additional saturation by
tive 1-hour-averagedzonal and meridional velocity profiles. convectiveinstabilitiesor strongnonlineardissipation,as proClearly, the analysis of such data in the manner described posedby Weinstock[1982]. Indeed, it is probably reasonable
above is hopeless.However, there is strong evidenceof wave to assumethat all of the mechanismsthat contribute to gravsaturation both in the radar data and from other sources. As
ity wave saturationare operatingin the complexfield of largehad been noted by Balsley et al. [1983] and Vincent [1984]
amplitudegravity wavespresentat the summermesopause.
for winter data (seeFigure 37), the wind variability and specIt is interestingto note that the occurrenceof shearinstabiltral width were found to remain nearly constant or increase ities due to low-frequencymotions near the summer mesoslowlywith heightnear the summermesopause.
The rms hori- pause appears to provide corroborative evidencefor the apzontal velocityobtainedbetweenJune 24 and July 8, 1981, by parent saturation of inertio-gravity wave motions in the lower
Balsleyet al. increased
from ~40 m s-x at 80-km altitudeto stratospherevia shearinstability.Thus evidenceexistsof grav~65 m s-x near 92-km altitude.But this represents
an in- ity wave saturation via both shear and convectiveinstability
creaseof only ~60% comparedwith an increaseof ~ 350% throughout the middle atmosphere. Convective instabilities
expectedfor conservativewave motionsand impliessubstan- appear to dominate whenever high-frequency, quasi-twotial wave dissipationthroughoutthis region.
dimensional gravity wave motions are involved. Shear instaAdditional evidence of saturation is available from the NLC
bility, on the other hand, appearsto be significant,though not
observationssummarizedby Haurwitz and Fogle [1969]. Toexclusive, when low-frequency wave motions predominate.
getherwith indicationsof verticallypropagatinggravity waves Finally, estimatesof vertical diffusivity are more in line with
at larger scales,theseauthorsobservedKelvin-Helmholtz bil- the predictionsof nonlinear saturationtheory, suggestingthat
lows to occur in ~45% of NLC sightings,providing direct the effects of wave superpositionmust be included in any
completedescriptionof gravity wave saturation.
80
i
i
i
i
i
5.
i
76
ßNorth
74
ßVertical .
72
-
70
N
ß ß ß
66
64
62
60
58
7.5
10.0
Hour
12.5
15•-0
of
Maximum
17.5
20.0
22.5
Fig. 39. Phaseprogressions
of the east,north, and vertical velocities obtained with a 10-hour least squaresfit to the wave field in
Figure 38 [Fritts eta!., 1984a]. Other wave parametersare identified
in the text.
PROCESSES DETERMINING
GRAVITY
78
THE
WAVE SPECTRUM
A large number of processescontribute to the maintenance
and characteristicsof the observedgravity wave spectrumin
the middle atmosphere.The most obvious of these,perhaps,
are the distribution and relative energy of the various tropospheric sources,the filtering and attenuation of the gravity
wave spectrumdue to propagationin a variable environment,
and the evolution of the spectrumdue to nonlinear wave-wave
and wave-turbulenceinteractions.Of these,only the filtering
processes
appear to be reasonablywell understoodat the present time. But even here there remain many unknowns.
The principal sourcesof gravity waves in the lower atmosphereare thought to be topography, wind shear, and convection, though many others,such as geostrophicadjustmentand
_
304
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
frontal accelerations, contribute as well. Of these, wind shear
and topography are believedto be the most important [Einaudi et al., 1978/79; Lindzen, 1984]. These are also the source
mechanisms
that have received the most theoretical
attention.
Holloway, 1980; McComas and Miiller, 1981a, b]. Also, the
spectra of fluctuations observed throughout the oceans and
the atmosphere do appear to have many similarities when
averaged over suitably large time intervals, suggestingthat
nonlinear wave-wave interactions may act to restore an equilibrium spectrum through a transfer of energy to (from) less
(more) energeticcomponentsfollowing perturbations due to
sourcesand sinksof wave energy.
There seemsto be a major difficulty, however,in the application of nonlinear interaction theory to the atmosphere.This
arises as a result of the frequent occurrenceof gravity wave
motions that appear to propagateunaffectedby theseinteractions. Examplesof suchmotions include the propagation and
saturation of nearly monochromatic wave motions and the
apparent utility of ray-tracing theory in caseswhere it has
been applied. These observationsimply relatively small interaction rates in preciselythose caseswhere the nonlinear interaction theory suggeststhat they should be very rapid. Thus it
is important to delineatethose situationsin which nonlinear
interactionsdo play an important role in the modificationof
the gravity wave spectrumin order that their effectsin the
middle atmospheremay be anticipated.
There remain, however, major uncertainties concerning the
distributionsof wavelength,phasespeed,and energyas well as
the temporal and spatial variability of theseexcitationmechanisms. It is probably reasonableto assumethat phase speed
distributions for topographic and shear-generatedgravity
waves are centered about zero and near tropospheric and
lower stratosphericwind speeds,respectively,based on observations and theoretical considerations.The phase speeddistribution for gravity waves generated by convectiveactivity is
likely to be very broad and centerednear troposphericwind
speeds.It is more difficult to anticipatethe dominant horizontal wavelengths associatedwith a particular source mechanism, becausethis will depend on the detailed nature of the
excitation. However, characteristicscalesof topography and
troposphericwind shears,togetherwith gravity wave observations, suggest horizontal wavelengths in the range of
• 10-1000 km. The sites of major topographic gravity wave
excitation are well known, but the temporal variation of the
energy and momentum fluxes due to these sourcesis not.
Likewise, the temporal and spatial variability of sheargeneratedand convectivelygeneratedgravity wavesis poorly
understoodat present.Together, these uncertaintiespreclude
a detailed descriptionof the spectrum of gravity waves entering the middle atmosphere.
The processescontributing to gravity wave filtering and
attenuationin the middle atmospherehave beenthe subjectof
numerousstudiesin recent years and are, for the most part,
This paper has provideda reviewof our currenttheoretical
and observationalunderstandingof gravity wave saturation in
the middle atmosphere.A brief history of the gravity wave and
middle atmospherestudiesleading to the recognitionof the
role of gravity waves in middle atmosphere processeswas
given in the introduction.The theoreticaland modelingstudies that have addressedvarious processesand effectsof satu-
well
ration
known.
These
include
saturation
and critical-level
ab-
sorption, radiative damping, diffusion and heat conduction,
and the reflectionof wave energyat turning levels.One factor
that may be important in this regard is the transientnature of
the environment through which an individual wave propagates.This is due to both quasi-lineareffectsand the presence
of other wave motions and may produce substantial departures from the behavior expectedon the basisof steady,linear
theory I-Fritts, 1982; Broutman,1982; Dunkerton,1982b; Coy,
1983; Fritts and Dunkerton, 1984]. But despiteour knowledge
of thesefiltering processes,
it is not possibleto anticipatetheir
effectsunlessthe gravity wave parametersare known.
Nonlinear interactions among internal gravity waves have
been recognizedto occur for over 2 decades.The case most
often considered is the resonant triad interaction
for which the
frequenciesand wave numbers of thesewaves add identically
to zero. This description of wave-wave interactions is only
accurate, however, if the wave amplitudes and interaction
rates are small. When the amplitudes are not small, offresonant interactions, which can be thought of as involving
additional, slowly interacting wave motions, may becomeimportant. The three resonanttriad interactionsthat have been
identified as being important in geophysicalfluid dynamics
are induced diffusion, elastic scattering, and the parametric
subharmonic instability I-McCornas and Bretherton, 1977].
Each of these represents a relatively efficient interaction
among waves in particular regionsof the gravity wave spectrum. Such interactions are thought to be responsiblefor the
maintenance of the observed"universal" gravity wave spectrum in the oceansand the atmosphere[Garrett and Munk,
1972, 1975; Van Zandt, 1982]. Indeed, a number of authors
have calculated high interaction rates due to relatively small
departures from an equilibrium spectrum [McCornas, 1977;
6.
SUMMARY AND CONCLUSIONS
were described
in some detail
in sections 2 and 3. Ob-
servationalstudiesrelating to saturation and the character of
the gravity wave spectrumwere reviewedin section4. Finally,
thoseprocesses
that are likely to affectthe maintenanceand
evolution of the gravity wave spectrumwere discussedbriefly
in section 5.
The linear saturation theory for two-dimensional monochromaticgravity waveswas developedin somedetail because
it providesthe simplestway in which the causesand effectsof
saturation can be understood. It was shown that saturation,
which was assumedto occur wherever the total lapse rate
becomessuperadiabatic,can result either from the growth of
wave amplitude with height or from the approachto a critical
level (where the intrinsic wave frequency vanishes).The assumedlimit on wave amplitude above the level at which saturation commenceswas found to cause a divergenceof the
momentum flux due to the wave motion and a corresponding
accelerationof the mean flow toward the horizontal phase
speed of the wave. It is this induced mean flow acceleration
that is now thought to provide the necessarydrag to balance
the momentum budget of the middle atmosphere and to
permit the reversalsof the mean zonal wind shear and the
mean meridional temperature gradient in the mesosphere.
Gravity wave drag may also produce substantialdecelerations
in the stratosphere,contributing to the maintenanceof weak
westerlymean flows at lower levels.A secondeffectof gravity
wave saturation is the generation of that level of turbulence
neededto dissipatethe gravity wave and maintain saturation
amplitudes.
While linear saturation theory has revealed two principal
effectsof gravity wave saturation, induced drag and turbulent
diffusion, it failed to address some of the more important
aspectsof saturationin the middle atmosphere.Consequently,
FRITTS: GRAVITY WAVE SATURATION IN THE MIDDLE ATMOSPHERE
a number
of studies have examined
various
extensions
and
implicationsof the linear saturationtheory. Those aspectsof
saturationnot addressedby linear theory that may be important in the atmosphereinclude the transport of heat and constituents through induced wave and turbulence fluxes, wave
reflectionand self-accelerationdue to quasi-linearmean flow
accelerations,wave scatteringdue to localized saturation,nonlinear interactionsamong a broad spectrumof wave motions,
and nonzonal drag and diffusion causedby filtering and attenuation of the gravity wave spectrum. Those studies addressingthe effectsof gravity wave saturation in the middle
atmospherehave shown that the saturation of even very crude
gravity wave spectracan accountfor the significantfeaturesof
the observed mean circulation. Nevertheless, there remain
many aspectsof the saturation problem that are poorly understood at presentand require additional study.
Observations of gravity wave structure, distribution, and
variability using a variety of techniqueshave revealed that a
wide range of wave motions participate in saturation throughout the middle atmosphere.Observed gravity wave scalesinclude horizontal wavelengthsof -• 10-1000 km, vertical wave-
lengthsof •<1-20 km, horizontalphasespeeds
of •<100m s- x,
and intrinsic frequenciesfrom f to •N. Other observationshave
shown the gravity wave spectrumin the middle atmosphereto
exhibit pronouncedlatitudinal, seasonal,and temporal variability. Preliminary studieshave suggestedthat both convective
and dynamical instabilitiesplay a role in gravity wave saturation, with convective instabilities predominant for highfrequency motions and both instabilities possible for lowfrequency motions. There is also evidence that saturation
often involvesa broad spectrumof wave motions.Both spectral energy measurementsand momentum flux measurements
have implied that decelerationis dominated by gravity wave
motions with relatively high frequenciesand small horizontal
scales.
305
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Finally, a major observationallimitation at this time is our
relativelyscantyknowledgeof overall gravity wave morphology, includingsourcedistributions,spectra,parameters,energy
levels,and those processes
that act to alter the gravity wave
spectrumthroughout the atmosphere.Clearly, more quantitative descriptionsof the gravity wave spectrumand its distribution and variability are neededbeforesubstantialadditional
progresscan be made in reconcilingtheory and observations
to provide a more complete understandingof the effectsof
gravity wave saturation in the middle atmosphere.Specific
recommendationsfor additional study can be found in the
workshoprecommendationsby Fritts et al. [1984b].
Acknowledgments.The author is grateful to B. B. Balsley, T. J.
Dunkerton, C. R. Philbrick, M. R. Schoeberl,T. E. Van Zandt, R. A.
Vincent, J. Weinstock, and an anonymousreviewer for numerous
helpful discussions
and comments.This work was sponsoredby the
Air Force Office of ScientificResearch(AFSC) under grant AFOSR82-0125 and the GeophysicalInstitute.
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(ReceivedOctober 27, 1983;
acceptedJanuary 4, 1984.)